REESE    LIBRARY  - 

OF   THE 

UNIVERSITY  OF  CALIFORNIA. 


Received 
Accessions  No. 


Shelf  No. 


JOHN  WILEY  &  SONS, 

Astor   Place,   New  York, 
PUBLISH: 

THE  RAILROAD  SPIBAL. 

The  Theory  of  the  Compound  Transition  Curve  reduced 
to  Practical  Formulae  and  Rules  for  Application  in  Field 
Work,  with  Complete  Tables  of  Deflections  and  Ordinates 
for  five  hundred  Spirals.  By  Wm.  H.  Seaiies,  C.E.,  author 
of  "Field  Engineering,"  Member  of  Am.  Soc.  of  C.  E. 

Pocket-book  form $1  50 

"  It  should  have  a  place  in  the  library  of  every  Civil  Engineer 
in  the  world."— Rail  way  ~4ye. 
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A  HAND-BOOK  of  the  Theory  and  Practice  of  RAIL- 
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and  containing  a  large  number  of  Useful  Tables,  Original 
and  Selected.  By  Wm.  H.  Searles,  C.E.,  late  Prof,  of 
Geodesy  at  Rensselaer  Polytechnic  Inst.,  Troy.  This  vol- 
ume contains  many  short  and  unique  methods  of  Laying 
Out,  Locating,  and  Constructing  Compound  Curves,  Side 
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AND 

EXPLORERS'   GUIDE. 

ESPECIALLY  ADAPTED  TO  THE  USE  OF 
RAILROAD  ENGINEERS 

ON 

LOCATION  AND   CONSTRUCTION, 

AND  TO  THE  NEEDS  OP  THE  EXPLORER  IN  MAKING 

EXPLORATORY   SURVEYS. 

BY 

H.  C.  GODWIN. 


NEW  YORK: 

JOHN    WILEY    &    SONS, 

15  ASTOR  PLACE. 

1890. 


Copyright,  1890, 

BY 
JOHN  WILEY  &  SONS. 


DRUMMOND  &  NEF,  FERRIS  BROS., 

Electrotype™,  Printers, 

1  to  7  Hague  Street,  s^  Pearl  Street, 
New  York.  New  York. 


PEEFACE. 


I  AM  publishing  the  following  notes  because  I  think  they 
may  possibly  supply  the  want  of  a  Field-book, — a  want  which 
I  have  often  felt  myself  and  have  often  heard  expressed — 
which,  while  avoiding  as  much  as  possible  the  intricacies 
of  mathematics,  would  be  of  more  general  application  than 
any  of  the  books  of  this  class  which  I  have  as  yet  come 
across. 

The  Railroad  engineer  is  rarely  an  expert  mathematician  : 
in  fact  it  has  always  seemed  to  me  that  the  time  which  must 
necessarily  be  spent  by  him  in  attaining  mathematical  pro- 
ficiency might  be  very  much  better  employed  in  reading  up 
some  of  the  more  practical  subjects  of  his  profession.  Bear- 
ing this  in  mind,  I  have  •  endeavored  to  strip  the  following 
pages  of  all  unnecessary  mathematical  deductions,  making  it 
mainly  my  object  to  give  the  results  deduced,  and  yet  at  the 
same  time  giving  sufficient  explanation  to  enable  any  one  pos- 
sessed of  the  ordinary  smattering  of  mathematics  and  me- 
chanics to  deduce  the  same  results  for  himself. 

I  have  avoided  the  insertion  of  Logarithmic  Tables.  I  am 
well  aware  that  to  some  this  will  appear  a  serious  omission; 
but  considering  that  this  is  merely  a  Field-book,  and  not  a 
work  to  be  consulted  in  cases  where  accuracy  in  the  6th  figure 
is  usually  essential,  I  have  deemed  that  the  exclusion  of  the 
hundred  pages  or  so  which  this  omission  permits,  amply  com- 
pensates for  the  few  seconds  of  additional  labor  which  the 
lack  of  them  may  occasionally  involve,  ^peaking  for  myself, 
as  regards  Railroad  work,  I  must  say  that  for  one  time  that  I 
work  by  logarithms  I  work  a  hundred  times  by  "  naturals  ;" 
and  I  know  that  most  engineers  would  bear  similar  testimony. 

In  the  Astronomical  problems  in  the  latter  part  of  the  book, 
considerable  labor  may,  of  course,  be  saved  by  the  use  of 

iii 


CONTENTS. 


BAILBOAD  LOCATION. 

GENERAL  CONSIDERATIONS. 

SEC.  PAGE 

1.  Conditions  of  Economical  Location 1 

2.  Train  Resistances 2 

3.  Rolling  Resistance 2 

4.  Resistance  due  to  Oscillation  and  Concussion 3 

5.  Atmospheric  Resistance 3 

6.  Resistance  due  to  Curvature 4 

7.  Resistance  due  to  Gravity 4 

8.  Diagram  of  Resistance .  5 

Limiting  Velocity  on  any  Grade , 8 

9.  Propelling  Force  of  Locomotive 8 

Coefficient  of  Adhesion 8 

Sliding  Friction : 8 

Limiting  I.H.P 9 

Weight  on  Driving-wheels 9 

Grate-surface 9 

10.  Diagram  of  Propelling  Force 10 

Limiting  Speed  for  any  given  I.H.P 10 

Internal  Frictional  Resistances.  11 

Back-pressure,  Wire-drawing,  etc 11 

11.  Diameters  of  Driving-wheels 11 

12.  I.H.P.  required  at  any  given  speed 12 

Most  Economical  Speed 12 

Limiting  Grade 13 

13.  Weight  of  Locomotives  and  Rolling-stock 13 

14.  Resistance  due  to  Inertia 13 

Rotative  Energy  of  the  Wheels. .  <, 14 

15.  Resistance  caused  by  Application  of  Brakes 14 

Automatic  Brakes  15 

Hand  Brakes 15 

16.  Initial  Velocity 15 

17.  Height  corresponding  to  Velocity 16 

Table  of  Heights  corresponding  to  Velocity 17 

18.  Assumption  of  Mean  Resistance  and  Mean  Propelling  Force..   .  17 

19.  Graphic  Method  of  solving  Dynamical  Problems 17 

20.  Examples 18 

V 


VI  CONTENTS. 


SEC.  PAGE 

21.  Rise  and  Fall 19 

Profile  of  Velocities 20 

22. 


Effects  of  Rise  and  Fall . 


24.  Maximum  Grade 21 

25.  Economy  of  Locomotive 22 

26.  Compensation  for  Curvature 22 

27.  Compensation  for  Brakes 23 

28.  Broken  Grades 23 

Momentum  Grades - 23 

29.  Danger  of  breaking  Train  and  Derailment 24 

30.  Work  done  on  Grades 24 

31.  Pusher-grades    26 

Table  of  Pusher-grades  26 

32.  Maximum  Curvature 26 

Safe  Speed  on  Curves  26 

33.  Short  Tangents 27 

Location  of  Curves 27 

34.  Table  of  Work  done  against  Resistances 28 

COST  OF  OPERATING. 

35.  Cost  of  Work  done  against  Resistance  28 

36.  Cost  per  Train-mile „ 28 

37.  Economy  of  Construction 29 

38.  Cost  of  Operating  Pusher-grades  30 

39.  To  test  Relative  Cost  of  Various  Routes 30 

40.  Effect  of  Alterations  in  Alignment 30 

41.  To  estimate  Effect  of  Ditto 31 

RECEIPTS. 

42.  Deviating  to  catch  Way-business  31 

COST  OF  CONSTRUCTION. 

43.  Average  Cost  of  Track 30 

Average  Cost  of  General  Construction 32 

Average  Cost  of  One  Mile  of  Track 33 

Cost  of  Trestle-work,  Trusses,  Tunnels,  etc 33 

INSTRUMENTS. 

44.  Transit  34 

Adjustments 34 

45.  Remarks 36 

46.  Stadia 39 

47.  Compass 42 

Adjustments 42 

Remarks 42 

48.  Magnetic  Variation 43 

Chart  of  Magnetic  Variation 44 

49.  Dumpy  Level 45 

50.  Y  Level 45 


CONTEXTS.  Vll 

SEC.  PAGE 

51.  Correction  for  Curvature  and  Refraction 46 

52.  Hand  Level 47 

THE  SURVEY. 

53.  Reconnoissance  and  Preliminary  Surveys 48 

Running  the  Line  to  Grade 49 

Table  of  Grades  and  Grade  Angles  50 

54.  Transit  Work 51 

55.  Latitudes  and  Departures 52 

56.  Azimuth  Observations 54 

57.  A.  Maximum  Elongation  of  Polaris  55 

B.  Observation  of  y  Cassiopeia  and  Polaris 56 

C.  Observation  of  Alioth  and  Polaris 57 

58.  Convergence  of  Meridians 58 

59.  Simple  Triangulations 60 

Offsetting  the  Transit-line 61 

60.  Levelling 61 

Precision  of  a  Line  of  Levels  ...  62 

61.  Taking  Topography 62 

62.  Contour  Lines 64 

Locating  by  means  of  Contour  Lines 64 

63.  Levels  and  Curvature 66 

64.  Equations 66 

65.  Value  of  Topography 67 

66.  Tangents  and  Curves  68 

67.  Selection  of  Curves  by  Eye 1 68 

68.  Balance  of  Cuts  and  Fills 69 

69.  Establishing  the  Grades 69 

Rough  Estimation  of  Grading 69 

70.  Estimating  by  Centre  Heights 71 

CURVES. 

71.  Radius  and  Degree  of  Curves 71 

72.  Corrections  for  50- foot  Chords 72 

73.  Length  of  Curves , 74 

74.  Nomenclature  and  Symbols 74 

75.  Fundamental  Formulae 75 

PROBLEMS  IN  SIMPLE  CURVES. 

76.  To  lay  out  a  curve  by  deflection  angles: 78 

To  find  corrected  length  of  any  sub-chord 78 

Example 78 

77.  To  locate  a  curve  when  the  apex  is  inaccessible 81 

78.  To  locate  a  curve  by  offsets  from  a  tangent 82 

Ditto  if  the  apex,  P.C.,  etc.,  are  inaccessible 83 

79.  To  locate  a  curve  by  offsets  from  the  chords  produced 85 

80.  To  locate  a  curve  by  ordinates  from  a  long  chord 87 

Example 87 

Ditto  by  mid-ordinates «8 


Vlll  COKTEKTS. 

SEC.  PAGK 

81.  To  pass  a  curve  through  a  fixed  point,  /  being  given 89 

82.  To  run  a  tangent  from  a  curve  to  any  fixed  point so 

83.  To  connect  two  curves  by  a  tangent 90 

84.  Given  a  curve  joining  two  tangents,  to  change  the  P.O.  so  that 

the  curve  may  end  in  a  parallel  tangent 91 

85.  To  transfer  a  curve  both  at  its  P.O.  and  P.T.  to  parallel  tangents.    92 

86.  Given  a  curve  joining  two  tangents,  to  change  R  and  the  P.O.  so 

that  the  new  curve  may  end  in  a  parallel  tangent  at  a 
point  opposite  to  the  original  P.T 93 

87.  Given  a  curve,  to  find  R  of  another  curve,  which,  from  the  same 

P.O.,  will  end  in  a  parallel  tangent 93 

88.  Given  a  curve  joining  two  tangents,  to  change  R  and  the  P.O.  so 

that  the  curve  may  end  in  the  same  P.T.,  but  with  a  change 
iu  direction 93 

COMPOUND  CURVES. 

89.  Locating  compound  curves 94 

90.  To  locate  a  C.C.  when  the  P.C.C.  is  inaccessible 94 

91.  Given  a  simple  curve  ending  in  a  tangent,  to  connect  it  with  a 

parallel  tangent  by  means  of  another  curve 95 

92.  To  connect  a  curve  with  a  tangent  by  means  of  another  curve 

of  given  radius 95 

93.  Given  a  C.C.  ending  in  a  tangent,  to  change  the  P.C.C.  so  that 

the  terminal  curve  may  end  in  a  given  parallel  tangent, 
without  changing  jts  radius 97 

94.  To  connect  two  curves  already  located  by  means  of  another 

curve  of  given  radius 98 

95.  To  locate  any  portion  of  a  C.C.  from  any  station  on  the  curve. .    99 

TRANSITION  CURVES. 

96.  Advantages  of  Transition  Curves 100 

97.  Method     1 100 

98.  Method   II 104 

99.  Method  III 105 

100.  Vertical  Curves . .  107 


CONSTEUCTION. 

101.  Division  of  the  Subject 109 

A.    SETTING  OUT  WORK. 

102.  Clearing  Right  of  Way,  etc 109 

103.  Location  of  Culverts,  etc 109 

104.  System  of  Drainage.— Ditches  110 

105.  Checking  Benchmarks  and  Alignment Ill 

106.  Cross-sectioning Ill 

Setting  Slope-stakes 112 

Points  at  which  Cross-sections  should  be  taken 114 


CONTENTS,  ix 


SEC.  PAGE 

107.  Reference  Points 115 

108.  Staking  out  Borrow-pits ; 115 

109.  Staking  out  Foundation-pits  for  Culverts  115 

110.  Setting  out  Bridge  Foundations 116 

111.  Setting  out  Trestlework 117 

112.  Setting  out  Tunnels 118 

113.  Giving  "Grade"  and  centres 120 

Shrinkage  and  Increase 121 

114.  Difference  of  Elevation  on  Curves 121 

Effect  on  the  Dump  and  on  Trestles 123 

Increase  in  Gauge  on  Curves 124 

115.  Inspecting  the  Grading 124 

116.  Running  Track-centres  and  setting  Ballast-stakes 125 

117.  Permanent  Reference-points  125 

118.  Turnouts  and  Crossings 125 

119.  Locating  by  Offsets 127 

120.  Example 129 

121.  Turnouts  and  Crossings  on  Curves 129 

122.  Curving  Rails 132 

123.  Expansion  of  Rails 132 

B.  THE  ESTIMATING  OF  LABOR  AND  MATERIAL. 

124.  The  Cost  of  Earthwork  and  Rockwork  removed  by  Carts 133 

125.  Ditto,  by  other  means. 136 

126.  Overhaul 136 

127.  The  Calculation  of  Earthwork , 137 

Areas  of  Cross-sections 139 

128.  The  Pyramid,  Wedge,  and  Prismoid 139 

129.  The  Prismoidal  Formula 140 

130.  The  Method  of  Average  End-areas  143 

Prismoidal  Corrections ...   143 

131.  The  Method  of  Equivalent  Level  Sections 146 

132.  The  Method  of  Centre-heights - 147 

133.  Earth-work  Tables 147 

134.  Correction  for  Curvature 148 

135.  Contents  of  the  Toe  of  a  Dump 149 

136.  General  152 

137.  Timber- work 152 

Table  of  Board  Measure 151 

Fractions  of  an  Inch  in  Decimals  of  a  Foot 153 

138  Iron-work * 153 

Weight  of  Bolts,  Nuts,  and  Bars 153, 154 

Railroad  Spikes 154 

Angle-bars  and  Bolts  per  mile 155 

Weight  of  Rails  per  mile 155 

139.  Ballast  and  Ties  per  mile 156 


CONTENTS. 


EXPLOEATOKY  SURVEYING. 

SEC.  PAGE 

140.  Introduction 157 

INSTRUMENTS. 

141.  The  Sextant.— Adjustments,  etc 157 

142.  Use  of  the  Sextant 159 

Parallax 159 

143.  Supplementary  Arc 161 

144.  Observing  Horizontal  Angles 161 

145.  Eliminating  Instrumental  Errors 162 

146.  The  Artificial  Horizon 162 

147.  The  Chronometer.  164 

148.  Barometers 165 

149.  Barometric  Formulae 166 

150.  Reduction  of  Errors  of  Gradient 168 

151.  Taking  Readings 169 

152.  Diurnal  and  Annual  Gradient 169 

153.  The  Cistern  Barometer 170 

154.  To  fill  a  barometer 170 

155.  Reading  the  barometer 171 

156.  Cleaning  the  barometer 171 

157.  The  Aneroid  Barometer 172 

158.  Elevation  Scales 173 

EXPLORATORY  SURVEYS. 

159.  Division  of  the  Subject 174 

160.  To  find  the  distance  apart,  etc.,  of  two  inaccessible  points 175 

161.  The  "  Three-point  Problem" 176 

162.  Positions  fixed  by  bearings 178 

163.  Positions  fixed  by  intersection 178 

164.  Obtaining  Heights  of  Mountains  trigonometrically 178 

165.  Refraction  of  the  Air 180 

166.  Reciprocal  Angles 180 

167.  By  Depression  of  the  Sea  Horizon 181 

168.  Observing  Altitudes  and  Depressions 181 

With  a  Sextant  and  Artificial  Horizon 181 

With  a  Transit * 182 

169.  Measurement  of  a  Base 182 

Correction  for  Temperature 182 

Reduction  to  Sea-level 183 

170.  Example  of  Triangulating  on  Exploratory  Surveys 183 

171.  To  measure  a  horizontal  angle  without  an  instrument 184 

172.  To  measure  a  vertical  angle  without  an  instrument 185 

173.  Measurement  of  Distance  by  Sound 185 

174.  Measurement  of  Time  by  Vibrations 185 

175.  Direct  Measurement  and  Compass  Courses 186 

Odometers  and  Pedometers 186 

Estimating  the  Rate  of  Progress 186 


CONTEXTS.  XI 


SEC.  PAGE 

176.  Astronomical  Observations 187 

177.  Solar  Time....  .187 

178.  Equation  of  Time -. 188 

179.  Sidereal  Time 189 

180.  Right  Ascension  and  Declination 189 

181.  Correcting  for  Longitude,  etc 190 

182.  Hour-angle 191 

183.  Examples 192 

184.  Refraction 194 

185.  Parallax 195 

186.  Correcting  for  Semi-diameter 197 

Augmentation 197 

187.  Dip 198 

188.  Summary  of  Corrections 198 

189.  Latitude.— By  Meridian  Altitudes 199 

190.  Remarks 201 

191 .  By  Transits  across  the  Prime  Vertical 202 

192.  By  an  Altitude  out  of  the  Meridian 203 

193.  By  Double  Altitudes 205 

194.  By  an  Altitude  of  Polaris  at  any  time  206 

195.  Longitude.— Local  Time,  by  an  Altitude  of  a  Star '-06 

196.  Local  Time,  by  Equal  Altitudes  of  a  Star  208 

197.  Local  Time,  with  a  Transit 208 

198.  By  Lunar  Culminations 209 

199.  By  Lunar  Distances 210 

200.  By  Jupiter's  Satellites 214 

201.  To  test  the  chronometer  rate \ 214 

202.  To  set  the  transit  in  the  meridian , 214 

203.  Interpolation  by  Successive  Differences 215 

204.  "  Accidental  Error" 216 

205.  Influence  of  Spheroidal  Form  of  the  Earth 218 

206.  Figure  of  the  Earth 218 

207.  Conversion  of  Angular  Measure  into  Distance  and  vice  versd 219 

208.  Given  the  lat.  and  long,  of  two  places,  to  find  their  distance 

apart,  etc  220 

209.  To  find  the  radius  of  a  circle  of  latitude 221 

210.  Offsets  to  a  Parallel  of  Latitude 221 

211.  Development  of  a  Spherical  Surface 221 

212.  Example 222 

213.  Star  Map 225 

Star  Tables .-. 226,  227 


MISCELLANEOUS. 


214.  The  Horse-power  of  Falling  Water 

215.  To  gauge  a  stream  roughly , 

216.  Sustaining  Power  of  Wooden  Piles 

217.  Supporting  Power  of  Various  Materials. . 


Xll  CONTENTS. 


SEC.  PAGE 

218.  Transverse  Strength  of  Rectangular  Beams 229 

219.  Natural  Slopes  of  Earth 230 

220.  Weight  of  Earths,  Rocks,  etc.,  per  cubic  yard , 230 

221.  Weight  of  Timber  and  Metals  per  cubic  foot 231 

222.  Mortar,  Cement,  and  Concrete 231 

223.  Notes  on  Timber.— Selection  of  Trees 231 

224.  Defects  of  Timber , 232 

225.  Felling  Timber 233 

226.  Seasoning  and  preserving  Timber 233 

227.  Decay  of  Timber 231 

228.  Tests  for  Steel  and  Iron 234 

229.  Strength  of  Rope.— Manilla,  Iron  and  Cast  Steel 235 

230.  Properties  of  the  Circle 236 

231.  Trigonometry.— Plane 237 

232.  General  Equations 240 

233.  Spherical 241 

234.  Measures  of  Length  and  Surface  243 

235.  Measures  of  Weight  and  Capacity 244 


APPENDIX. 

TABLES. 
Table          I.  Radii  of  Curves 252 

"  II.  Tangents  and  Externals  to  a  1°  Curve 255 

44  III.  Tangential  Offsets  at  100  feet 259 

IV.  Mid-ordinates  to  100-foot  Chords 259 

44  V.  Long  Chords 260 

41  VI.  Mid-ordinates  to  Long  Chords 263 

44         VH.  Minutes  in  Decimals  of  a  Degree 264 

44        VIII.  Squares,  Cubes,  Square  and  Cube  Roots 265 

IX.  Logarithms  of  Numbers.— 1  to  1000 282 

44  X.  Natural  Sines  and  Cosines 285 

41  XI.         '4       Secants  and  Cosecants 294 

"          XII.         4'       Tangents  and  Cotangents 309 

44         XIII.         '4       Versines  and  Exsecants 321 

44        XIV.  Cubic  Yards  per  100  feet,  in  terms  of  Centre-height. . . .  345 
XV.  Cubic  Yards  per  100  feet,  in  terms  of  Sect-:.>iial  Area. . .  350 
44        XVI.  Mutual  Conversion  of  Feet  and  Inches  into  Meters  and 

Centimeters 354 

44       XVII.  Mutual  Conversion  of  Miles  and  Kilometers 355 

"     XVIII.  Length  of  1'  arcs  of  Latitude  and  Longitude 355 

44        XIX.  Mutual  Conversion  of  Mean  and  Sidereal  Time , 356 

*4         XX.  Mutual  Conversion  of  Time  and  Degrees 358 


ERRATA. 

Page    4,  line  10.     For  .27  read  3.7 
"    24,    "    11.      "      12     "    18 
'•    36,    "    16.     After  them  wi*er<  say  A 
"    36,    "    24.     For  1.59  read  1.84 
"    55,  fo«0m  fone.     -For  ZP  read  WP 
"    72,  2tn«  3.    For  radius  read  degree,  and  for  degree  read 

radius 

"    78,  line  17.     For  abc  read  acb 
"  117,     ^Equation  in  line  8  should  be 

Be  =  R  f  sin  POM  -  ^- 
ct  128,  fone  3.     Far  AB  read  AD 


PART  I. 
BAILROAD  LOCATION. 


GENERAL   CONSIDERATIONS, 

1.  IN  the  early  days  of  Railroad  Building,  the  Locating  En- 
gineer was  forced  to  rely  mainly  on  his  individual  ability, 
trusting  principally  to  the  correctness  of  his  eye  to  detect  the 
most  suitable  route,  guided  only  by  the  very  limited  experience 
of  others  and  his  own  common-sense.  The  man  who  worked 
his  party  the  hardest,  and  covered  most  ground  in  the  day, 
was  in  those  days,  unless  any  very  obvious  defects  were  visi- 
ble in  his  work,  too  often  looked  upon  as  the  best  locator. 
But  the  years  of  experience  which  have  followed  have  been 
years  of  experiment  also  ;  and  the  practice  of  Railroad  Loca- 
tion has  by  degrees  developed  into  a  science,  which,  though 
yet  far  from  perfect,  forms  a  most  important  part  of  a  Modern 
Engineering  Education. 

In  a  Field  book  of  this  sort,  it  is  impossible  to  do  more  than 
treat  rapidly  a  few  of  the  leading  questions  which  the  subject 
involves,  and  formulate,  where  possible,  rules  for  guidance  in 
the  field. 

A  knowledge  of  the  principles  of  Railroad  Location  must 
be  backed  up  by  experience  in  Railroad  Construction.  For, 
in  order  to  locate  well,  a  man  must  have  fairly  accurate  ideas 
of  the  suitability  and  cost  of  the  various  works  which  his  lo- 
^ation  involves.  The  best  location  for  a  certain  road  is  not 
that  which  enables  the  traffic  to  be  carried  on  with  the  least 
amount  of  work,  or  which  gives  the  lowest  Operating  Expenses, 
but  that  which,  in  a  given  time,  renders  the 

ReceiDtS  -  /°Peratin^        /Interest  on  Capital  spent  on\  _  p_nfit_ 
VExpensesJ        ( Construct.  Equipment,  etc .)  ~ 


2  KAILROAD   LOCATION. 

a  maximum.  Thus  we  see  that  more  or  less  accurate  esti- 
mates of  the  probable  Receipts  and  Operating  Expenses  are  of 
the  utmost  importance  before  starting  the  location  ;  and  it  is 
only  when  these  are  arrived  at  that  the  amount  which  we  are 
entitled  to  expend  on  construction  can  be  fixed. 

2.  Before  considering  the  Financial  side  of  the  question, 
however,  we  will  glance  hurriedly  over  some  of  the  principal 
Mechanical  Problems  which  occur  in  dealing  with  the  motion 
of  trains,  for,  without  some  slight  knowledge  of  Railroad 
Dynamics,  an  intelligent  application  of  the  Laws  of  Location 
is  impossible. 

TEAIN  EESISTANCES. 

The  Resistance  due  to  the  motion  of  a  train  on  a  straight 
level  track— excluding  for  the  present  the  Inertia  of  the  train — 
may  be  regarded  as  being  the  sum  of  the  three  following  com- 
ponents : 

3.  ROLLING  RESISTANCE,  which  is  composed  of  the 
frictional  resistance  at  the  journals  and  that  at  the  wheels  at  the 
points  of  contact  with  the  rails  :  these  two  may  for  ordinary 
purposes  be  classed  together  under  the  head  of  Rolling  Resist- 
ance.    Its  magnitude  depends  largely  upon  the  surface  bear- 
ing at  the  journals  ;  the  coefficient  of  friction  decreasing  as 
the  load  per  unit-surface  on  the  journals  increases,  so  that  the 
resistance  is  relatively  higher  in  the  case  of  Empty  Cars  than 
with  Loaded  ones  ;  being  at  ordinary  speeds  about  6  Ibs.  per 
ton  (2000  Ibs.)  of  weight  of  train  in  the  former  case,  while  with 
Passenger  Coaches  or  Loaded  Cars  it  only  amounts  to  about 
4  Ibs.     By  referring  to  the  Diagram  of  Resistances,  p.  6,  we 
see  that  at  the  point  of  starting  the  Rolling  Resistance  is  very 
high,  being  then  about  20  Ibs.  per  ton,  but  that  at  a  velocity  of 
about  ten  miles  per  hour  it  reaches  its  minimum  value,  and 
from  that  point  increases    constantly  by  a  trifling    amount 
through  the  successive  higher  velocities.     The  Initial  Resist- 
ance depends  largely  on  the  length  of  time  the  train  has  been 
standing,  a  stop  of  only  a  few  seconds  causing  a  resistance  of 
about  one  half  that  given  in  the  Diagram.     Since,  however, 
there  is  always  more  or  less  "  give  "  about  the  couplings,  no 
two  cars  at  the  same  instant  offer  their  maximum  resistance, 
the  front  end  of  a  long  train  being  well  under  way  before  any 
motion  at  all  is  transmitted  to  the  rear.     Thus  the  pull  on  tke 


KAILROAD   LOCATION.  3 

draw-bar  is  not  in  reality  so  excessive  as  it  at  first  appears ; 
for  if  we  take  the  whole  train  into  consideration,  the  resist- 
ance at  the  start  may  be  set  down  as  about  12  Ibs.  instead  of 
20  Ibs.  per  ton,  as  in  the  case  of  a  single  car. 

The  Line  of  Rolling  Resistance  starts  in  the  Diagram  from 
the  line  of  the  1  p.  c.  grade  ;  thus  indicating  that  a  train  left 
standing  with  the  brakes  off  on  this  grade,  is  just  on  the  point 
of  starting  on  its  own  account.  On  any  grade  lighter  than 
this,  a  train  will  usually  require  considerable  force  to  set  it  in 
motion.  By  increasing  the  diameters  of  the  wheels  we  slightly 
decrease  the  resistance  to  rolling. 

4.  RESISTANCE  DUE  TO  OSCILLATION  AND  CON- 
CUSSION.— The  amount  of  this  we  obtain  approximately  by 
assuming  that  it  equals  .005  Ib.  per  ton  at  1  mile  per  hour,  and 
increases  as  the  square  of  the  velocity.    Thus,  e.g.,  at  40  m.  p.  h. 
it  equals  8  Ibs.  per  ton.    The  longer  the  train,  however,  the  less 
this  resistance  amounts  to  per  ton,  for  each  car  is  more  or  less 
steadied  by  the  force  which  is  transmitted  through  it  to  the  ad- 
joining one;  thus  it  is  usually  much  more  considerable  in  the 
rear  than  in  the  centre  or  forward  end  of  the  train.     It  is  pro- 
duced in  a  great  measure  by  the  inequality  in  elevation  of  the 
two  rails  on  an  imperfect  track,  and  thus  is  often  found  to  dimin- 
ish on  curves  where  the  difference  in  elevation  of  the  rails  is 
not  exactly  suited  to  the  speed  at  which  the  train  is  travelling, 
since  it  is  then  subjected  to  a  lateral  thrust  which  prevents 
the  oscillations  being  so  great  as  they  otherwise  would  be. 

5.  ATMOSPHERIC  RESISTANCE.— This  is  due  to  two 
causes : 

(a)  The  opposition  offered  by  the  particles  of  air  in  the 
direct  path  of  the  engine,  while  being  thrust  forwards  and 
sideways  by  the  advancing  train,  together  with  the  "  suction" 
caused  by  the  rear  car  ;  and — 

(b)  The  frictional  resistance  of  the  air  against  the  surface  of 
the  train,  corresponding  to  the  "  skin  resistance  "  in  the  case 
of  ships.     The  former  (a)  amounts  to  about  0.3  Ib.  per  train 
running  through  still  air  at  a  velocity  of  1   mile  per  hour, 
and  increases  as  the  square  of  the  speed  :  thus,  e.g.,  at  40  m. 
p.  h.  it  amounts   to  about   480    Ibs.     Probably  in  ordinary 
trains  not  more  than  one  third  of  this  resistance  causes  addi- 
tional strain  on  the  draw-bar,  because  the  greater  part  of  it  is 
taken  and  overcome  by  the  engine  itself.     As  regards  the  latter 


4  EAILEOAD   LOCATION. 

resistance,  (b)  it  may  be  ascertained  with  tolerable  accuracy  by 
allowing  0.03  Ib.  per  car  at  a  speed  of  1  mile  per  hour,  and 
considering  it  to  increase  as  the  square  of  the  velocity.  Thus, 
if  we  have  a  train  composed  of  10  loaded  box-cars  (see  Sec.  13) 
hauled  by  an  engine  which,  together  with  its  tender,  weighs 
60  tons,  .the  total  atmospheric  resistance  in  Ibs.  at  40  m.  p.  h. 
—  480  -f-  480  =  960  Ibs.  (assuming  that  the  allowance  already 
given  for  the  engine  includes  the  surface  resistance  as  well) ; 
and  since  the  weight  of  the  train — inclusive  of  engine  and 
tender — equals  about  260  tons,  this  is  equivalent  to  about  3R  31 
Ibs.  per  ton  of  entire  train.  Suppose,  in  the  above  example, 
we  have  a  Head-wind  blowing  at  the  rate  of  20  m.  p.  h.,  we 
may  then  consider  the  atmospheric  resistance  as  being  that 
due  to  a  train  velocity  of  60  m.  p.  h.  But  if  this  wind  were 
blowing  in  the  same  direction  in  which  the  train  is  going, 
then  the  resistance  caused  by  it  would  be  equal  to  that  caused 
by  a  train  velocity  of  20  m.  p.  h.  in  still  air. 

A  Side-wind  adds  very  considerably  to  the  ordiuar}^ atmos- 
pheric resistances  by  increasing  the  frictional  resistance  at  the 
rails,  owing  to  the  flanges  of  the  wheels  being  pressed  against 
the  inner  side  of  the  leeward  rail. 

The  above  resistances  are  peculiar  to  all  trains  at  all  times  ; 
the  two  following,  however,  are  accidental,  and  dependent  on 
circumstances. 

6.  RESISTANCE  TO  CURVATURE.— The  many  causes 
which  combine  to  make  up  this  resistance,  and  the  share  which 
each  has  in  forming  the  result  as  a  whole,  have  been  but 
vaguely  determined  by  experiment:    it  is  known,  however, 
that    at    speeds    not    exceeding  about   5  miles  per  hour,  it 
amounts  to  about  2  Ibs.  per  ton  per  degree  of  curvature,  and 
that  it  decreases  as  the  speed  increases,  as  shown  in  Diagram 
I,  till  at  70  miles  per  hour  it  does  not  probably  amount  to 
more  than  i  Ib.  per  ton.     Thus,  e.g.,  on  a  5°  curve  it  amounts 
at  a  velocity  of  35  m.  p.  h.  to  about  2  Ibs.  per  ton. 

The  use  of  Transition  curves  (page  100)  is  found  to  decrease 
it  materially. 

7.  RESISTANCE  DUE  TO  GRAVITY.— This  resistance 
may  be  termed  a  "  mathematical  "  one,  whereas  the  previous 
ones  have  been  based  entirely  on  experiment ;  for  though  the 
coefficient  of  gravity  is  itself  a  quantity  derived  from  experi- 
ment, it  is  merely  the  ratio  of  the  inclined  component  AB 


RAILROAD  LOCATION". 


5 


(Fig.  1)  to  the  force  of  gravity  A  C,  which  enters  into  the 
question  ;  or,  what  is  the  same  thing,  the  ratio  of  ab  to  ac. 


But  since,  in  dealing  with  ordinary  inclines,  we  may  con- 
sider ac  =  cb,  we  may  say  that 

AB_ab 

AC~cb' 

so  that  the  resistance  caused  by  gravity  per  ton  (2000  Ibs.)  equals 
in  Ibs.  20  X  fate  per  cent  of  tlie  grade.  Thus  on  a  2.5  p.  c.  up- 
grade the  gravity  resistance  equals  50  Ibs.  per  ton. 


DIAGRAM  OF  RESISTANCES. 

8.  We  are  now  in  a  position  to  draw  the  Line  of  Resist- 
ance for  any  given  train  under  any  ordinary  conditions. 
This  line,  for  a  train  on  a  straight  level  track,  is  found  by 
setting-off  at  the  successive  velocities  the  sum  of  the  ordinates 
for  the  Resistances  given  in  Sections  3,  4,  and  5  ;  and  the  line 
representing  each  of  these  component  resistances  can  be  read- 
ily plotted  with  the  aid  of  the  information  already  given. 
Suppose,  however,  that  the  train  is  running  on  a  curve  of, 
say,  10°,  we  must  then  measure  the  respective  ordiuates  to  the 
resistance  line  for  the  10°  curve,  and  #dd  these  to  the  ordi- 
nates already  obtained.  We  then  get  the  Line  of  Total  Resist- 
ance on  a  10°  curve.  If  in  addition  to  the  10°  curve  we  have 
a  Jr  0.25  per  cent  grade,  we  have  simply  to  add  the  height 
given  on  the  diagram  for  this  grade  to  each  of  the  ordinates 
already  found,  in  order  to  obtain  the  Line  of  Resistance  for 
the  train  on  a  10°  curve  and  a  -f-0.25  p.  c.  grade.  If  the 
train  were  descending  the  grade,  it  would  be  necessary  to  sub- 
tract the  last  ordinate  instead  of  adding  it. 


DIAGRAM  I. 

TRAIN  RESISTANCES  IN  LBS.   PER  TON. 

Engine  and  Tender  weigh  60  tons.    10  Loaded  Box-Cars,  each  weighing  20  tons. 
SCALE,  1  inch  vert.  ^=  10  Ibs.  (6) 


DIAGRAM  II. 

PROPELLING  FORCE  OF  LOCOMOTIVE  IN  LBS.   PER  TON. 
Locomotive  500  I.  H.  P.  Engine  and  Tender  =  60  tons. 

f  =  0.2  10  Cars,  20  tons  each. 

SCALE,  1  inch  vert.  =  10  Ibs. 


8  RAILROAD   LOCATION. 

In  order  to  find  the  Limiting  Telocity  of  any  train  on  a 
certain  grade,  moving  solely  under  the  influence  of  gravity, 
we  have  only  to  find  the  point  of  intersection  of  the  line  of 
total  resistance,  for  a  level  track,  with  the  horizontal  line  cor- 
responding to  the  grade  in  question,  and  notice  the  velocity 
corresponding  to  this  point.  Thus  in  Diagram  I,  for  the  train 
there  given,  running  round  a  10°  curve  down  a  2  p.  c.  grade, 
the  limiting  velocity  will  be  about  63  in.  p.  h. 

9.  Next  comes  the  consideration  of  the  counteracting  force, 
namely : 

THE  PEOPELLING  FOKCE   OF   THE 
LOCOMOTIVE. 

The  Coefficient  of  Adhesion,  i.e.,  Static  friction,  between 
the  rails  and  the  driving-wheels  of  a  locomotive,  is  found  to 
be  much  the  same  at  all  speeds,  but  to  increase  rapidly  as  the 
load  per  unit-surface  increases.  It  varies  in  ordinary  Rail- 
road practice  from  about  0.33  when  sand  is  used  to  about  0.18 
when  the  rails  are  slippery.  Under  ordinary  circumstances 
the  maximum  Propelling  Force  of  a  Locomotive  may  be  con- 
sidered equal  to  one  fifth  the  weight  on  its  drivers,  assuming 
0.2  as  the  usual  working  coefficient  of  adhesion  ;  thus  varying 
from  about  a  ton  to  a  ton  and  a  half  per  driving-wheel,  ac- 
cording to  the  type  of  locomotive. 

If  on  starting  a  train  the  driving-wheels  are  allowed  to  slip 
on  the  rails,  the  friction  is  no  longer  Static  but  Sliding1,  the 
coefficient  of  which  equals  about  0.1,  decreasing  rapidly  as 
the  velocity  increases  ;  which  shows  the  fallacy  of  allowing 
the  wheels  to  slip.  The  part  of  the  rail,  however,  on  which  the 
slipping,  if  any,  has  taken  place  is  found,  if  the  engine  is 
reversed,  to  give  a  coefficient  of  adhesion  higher  than  else- 
where. 

Where  Two  or  more  pairs  of  wheels  are  coupled  together, 
the  adhesive  force  is,  of  course,  due  to  the  load  on  all  the 
wheels  coupled  to  the  driving-wheels. 

Now,  however  great  steam-producing  capacity  the  locomo- 
tive may  possess,  its  Propelling  Force  is  limited  by  the  coeffi- 
cient of  adhesion  ;  and  though  it  can  expend  its  full  power  in 
spinning  the  wheels  around,  the  portion  of  this  power  which 


RAILROAD   LOCATION. 

can  be  utilized  for  propelling  the  train   is   limited  by  the 
amount  expressed  in  Indicated  Horse-Power : 

I.  H.  P.  =  5.9  WfV, 

where  W=  total  weight  in  tons  (2000  Ibs.)  on  the  drivers, 
/  =  coefficient  of  adhesion, 
V  =  velocity  in  miles  per  hour. 

This  formula  allows  10  p.  c.  for  overcoming  the  Internal 
Resistances  in  the  engine  itself  (see  page  11).  The  friction 
at  the  journals  of  the  driving-wheels,  however,  is  not  included 
among  these,  but  is  allowed  for  in  the  ordinary  Rolling  Re- 
sistance already  dealt  with.  Thus  if  we  take  the  weight  on 
each  driving-wheel  as  6  tons,  and/=  0.2,  the  above  formula 
becomes 

I.  H.  P.  =  1NV (nearly), 

where  N=  the  number  of  driving-wheels. 

Thus,  e.g.,  if,  in  an  ordinary  locomotive  with  four  driving- 
wheels,  we  have  the  production  of  steam  equivalent  to  400 
I.  H.  P.,  we  see  that  it  is  unable  to  utilize  its  full  power  for 
propelling  purposes  until  it  attains  the  speed  of  about  14 
miles  per  hour,  at  which  point  an}r  slight  increase  in  pressure 
would  cause  the  wheels  to  slip.  Thus  up  to  a  certain  speed 
the  propelling  power  of  an  engine  is  limited  by  the  weight  on 
its  drivers,  but  remains  more  or  less  constant  until  that  speed 
is  attained,  after  which,  instead  of  being  limited  by  the  adhe- 
sion of  the  wheels,  it  is  mainly  a  question  of  the  steam-pro- 
ducing power  of  the  boiler, 

In  ordinary  practice,  1  square  foot  of  Grate-surface  is 
able,  at  ordinary  speeds,  to  maintain  the  production  of  steam 
equivalent  to  24  I.  H.  P. :  so  that  if  we  know  the  total  grate- 
surface  of  an  engine  and  the  load  on  its  drivers, — assuming  it 
to  be  tolerably  well-proportioned  in  its  various  parts, — we  can 
form  a  fair  idea  of  its  tractive  power.  The  usual  allowance 
of  grate-surface  varies  from  about  15  square  feet  in  Passenger 
Engines  to  double  this  amount  in  some  of  the  Heavy  Freight 
Engines  :  thus  the  power  of  an  ordinary  Passenger  Engine, 
when  working  under  ordinary  conditions,  equals  about  360 
I.  H.  P.,  and  in. the  case  of  a  heavy  Freight  Engine  about 
720  I.  H.  P.  Both  these  classes  of  engines  can,  and  often  do, 
maintain  very  much  higher  powers  than  these,  but  to  work 
very  considerably  above  them  over  a  long  run  is  a  severe  tax 
on  the  economy  of  the  engine. 


10  KAILROAD  LOCATION. 

DIAGRAM  OF  PROPELLING  FORCE. 

10.  In  order  to  ascertain  the  probable  effect  of  a  given  lo- 
comotive on  a  certain  train  on  various  grades  and  curves,  it  is 
best  to  draw  the  Line  of  Propelling  Force  of  the  Engine 
—i.e.,  the  Line  of  Tractive  Power  exerted  at  the  point  of  con- 
tact of  the  driving-wheels  with  the  rails— in  Ibs.  per  ton  (2000 
Ibs.),  of  the  weight  of  the  engine  and  train. 

Suppose,  as  in  Diagram  II,  we  wish  to  find  the  effect  of  a 
locomotive  capable  of  maintaining  a  working  power  of  500 
I.  H.  P.  having  four  drivers  with  6  tons  on  each;  and  let  the 
engine  with  its  tender  weigh  60  tons,  and  the  train  be  the 
same  as  that  for  which  the  Lines  of  Resistance  are  given  in 
Diagram  I,  namely,  10  loaded  box  cars,  each  weighing  20 
tons—/  being  taken  as  0.2.  We  then  have  a  fair  example  of 
the  working  of  a  Light  Freight  Engine. 

Draw  the  Line  of  Propelling  Force  as  follows : 

Make  OA  =  J™~~^     r-  =  36.9  Ibs.  per  ton. 

lot.  Weight  of  Train 

I  H   P 

Then  draw  Aa  =  -'^^^  =  17.6  miles  per  hour, 
o.y  w  j 

which  (according  to  Sec.  9)  gives  the  velocity  above  which 
slipping  cannot  occur.  Now  the  theoretic  curve  of  Propel- 
ling Force  will  be  a  hyperbola,  drawn  through  a  (AO  and 
OH  being  its  asymptotes).  This  curve  may  be  drawn  by  off- 
sets from  OA  thus  :  At  a  distance  along  OA  from  0  equal  to 
\OA,  the  offset  equals  lAa ;  at  a  distance  equal  to  \OA,  the 
offset  equals  2 Aa,  and  so  on ;  the  offset  varying  inversely  as 
its  perpendicular  distance  from  0.  Then  C,  the  point  of  in- 
tersection of  the  Line  of  Propelling  Force  with  the  Line  of 
Resistance,  gives  the  Limiting  Speed  at  which  the  engine 
can  haul  the  train,  under  the  conditions  for  which  the  line  of 
resistance  is  drawn, — in  this  case,  on  a  straight  level  track. 

Then,  taking  any  ordinate  such  as  NMPQ,  the  part  NM  in- 
cluded between  the  Line  of  Propelling  Force  and  the  Line  of 
Resistance  gives  that  portion  of  the  propelling  force  of  the 
engine  in  Ibs.  per  ton  (2000  Ibs.)  which  goes  to  overcome  the 
Inertia  of  the  train  at  the  speed  indicated. 

But  this  Line  of  Propelling  Force  assumes — as  we  men- 
tioned before— that  10  per  cent  of  the  I.  H.  P.  is  absorbed  in 


BAILED  AD   LOCATION.  11 

overcoming  the  Internal  Frictional  Resistances  of  the  en- 
gine itself — exclusive  of  the  resistance  at  the  journals — inde- 
pendent of  the  velocity.  At  low  speeds  this  allowance  is  con- 
siderably too  much,  but  at  high  velocities  it  is  insufficient ; 
for  ordinary  speeds,  however,  it  will  not  be  far  from  correct. 
The  journal-friction  forms  probably  about  one  third  of  the 
whole  :  the  friction  of  the  piston,  slide-valve,  valve-gear,  and 
cross-heads  also  contribute  considerably  to  the  total.  Very 
little  is  known  as  to  what  allowance  ought  to  be  made  to 
cover  these  resistances, — in  fact  it  is  so  much  a  matter  of  lu- 
brication and  mechanical  detail  that  no  general  formula  could 
be  applied, — but  undoubtedly  they  increase  with  the  velocity, 
and  are  higher  in  an  engine  hauling  a  heavy  train  than  in  an 
engine  running  light. 

Also  we  have  Back-pressure  of  the  steam  in  the  cylinders, 
Wire-drawing,  and  various  other  causes  entering  into  the 
question  at  high  speeds  which  also  tend  to  lessen  the  effective 
Horse-power. —See  Note  A,  Appendix. 

11.  Now  since  the  loss  of  power  due  to  these  causes  de- 
pends largely  on  the  rotary  velocity  of  the  Driving-wheels, 
in  the  case  of  two  engines  both  developing  the  same  I.  H.  P. 
at  the  same  speed, — the  cylinders  being  suitably-proportioned, 
— the  engine  with  the  larger  wheels  will  have  a  great  advan- 
tage over  the  other  at  high  speeds,  although  at  low  speeds  the 
engine  with  the  smaller  wheels  will  have  the  best  of  it.  At  low 
speeds — since  the  initial  pressure  in  the  cylinders  then  differs 
but  little  from  the  boiler-pressure  and  the  back-pressure  is 
practically  nothing — an  engine  with  several  small  drivers  will 
of  course  have  an  enormous  advantage  over  an  engine  of  the 
same  I.  H.  P.  with  only  a  single  pair  of  large  drivers  on  ac- 
count of  its  being  able  to  utilize  so  much  more  of  its  power, 
by  reason  of  its  higher  adhesive  qualities.  For  instance,  it 
would  probably  tax  the  engine  with  Jarge  drivers  severely  to 
start  a  train  which  the  other  engine  could  handle  with  ease; 
but  when  the  speed  reached,  say,  thirty  miles  per  hour,  the 
engine  with  the  large  drivers  could  work  it  much  more  easily 
and  economically  than  the  engine  with  the  small  ones.  Thus 
where  high  velocities  are  required, — whether  on  heavy  grades 
or  not,  provided  the  weight  on  the  drivers  is  sufficient, — if  the 
cylinders,  etc.,  are  suitably  proportioned,  the  wheels  of  large 
diameter  are  decidedly  the  best. 


12  RAILROAD  LOCATION. 

Mr.  Wellington  states  that  in  the  case  of  ordinary  Passenger 
Engines  and  trains  of  medium  length,  50  per  cent  of  the 

I.  H.  P.  is  consumed  in  the  locomotive  itself,  overcoming  its 
various   resistances — atmospheric,  rolling,  internal,  etc., — so 
that  only  one  half  of  the  Horse-power  produced  is  trans- 
mitted through  the  draw-bar. 

From  the  foregoing  it  appears  that  a  closer  approximation 
to  the  true  line  of  propelling  force  at  high  velocities  may  be 
found  by  drawing  it  as  shown  by  the  dotted  -line  in  Diagram 

II,  somewhat  below  the  theoretic  line  already  drawn.     The 
intersection  of  this  line  with  OH  (produced)  gives  the  maxi- 
mum speed  of  the  engine  if  unopposed  by  any  external  resist- 
ances,— i.e.,  if  running  free  as  a  stationary  engine, — 10  per 
cent  only  of  the  power  developed  being  absorbed  in  overcom- 
ing internal  resistances. 

It  must  be  remembered  that  the  Line  of  Propelling  Force 
shown  in  the  Diagram  is  at  all  points  the  maximum  which 
can  be  obtained  without  exceeding  the  I.  H.  P.  stated  ;  but  by 
taking  a  comparatively  low  value  of  /,  and  a  high  allowance 
for  the  internal  frictional  resistances  of  the  engine  at  low 
speeds,  we  obtain  by  the  method  given  probably  as  correct 
results  as  can  be  obtained  by  any  mathematical  process. 

12.  If  we  require  to  know  what  I.  H.  P.  an  Engine 
must  develop  to  haul  a  certain  train  at  a  given  velocity  V, 
we  can  find  it  at  once  theoretically  by  multiplying  the  total 
weight  of  the  engine  and  train  in  tons  (2000  Ibs.)  by  the  resist- 
ance in  Ibs.  per  ton  (taken  from  Diagram  I)  and  multiplying 
the  product  by  .003F(Fbeing  in  miles  per  hour).  Thus 
with  the  train  given  in  Diagram  II,  we  should  need  an  en- 
gine capable  of  developing  about  950  I.  H.  P.  in  order  to  haul 
it  at  a  speed  of  50  miles  per  hour.  The  I.  H.  P.  exerted  in- 
creases nearly  as  F3,  and  the  tractive  force  nearly  as  F2. 
The  total  amount  of  steam  used  theoretically,  on  a  run,  is 
nearly  proportional  to  F'2.  The  most  economical  speed,  as 
regards  fuel,  at  which  a  train  can  be  run — provided  the  en- 
gine is  of  a  power  suitable  to  the  weight  of  the  train — is  found 
by  experiment  to  be  about  18  miles  per  hour,  and  not,  as  might 
be  expected  from  Diagram  I,  at  about  8  miles  per  hour. 
This  is  due  mainly  to  the  saving  in  heat  owing  to  the  engine 
being  a  shorter  time  on  the  trip,  and  also  on  account  of  the 
smaller  effect  produced  by  variations  in  grade  at  the  higher 


BAILBQAD   LOCATION". 


13 


velocity.  To  ascertain  the  Limiting  Grade  which  it  is  possi- 
ble to  work,  we  find  from  the  diagrams  that  an  engine  and 
tender  weighing  together  60  tons,  with  24  tons  on  the  drivers, 
can  under  ordinary  conditions  just  make  head-way  up  a  12- 
per-cent  grade  ;  and  that  it  is  just  all  two  engines  of  the 
above  description  can  do  to  haul  a  passenger  coach  up  a  10- 
per-cent  grade. 

13.  The  following  may  be  taken  as  fair  examples  of  the 

WEIGHT  OF  AMERICAN  ROLLING-STOCK: 


Type. 

No.  of 
Drivers. 

Weight  in 
tons  on 
each 
Driver. 

Weight  in  tons, 
engine  and  tender, 
with  fuel  and 
water. 

Heavy  Passenger  Engine.  .  . 
Consolidation  Engine  
Decapod  Engine. 

4 
8 
10 

!« 

7 

55 

75 
95 

feet- 


(1  ton  ?=  2000  Ibs.) 

veight  10  tons.  ) 

"      20     "  f 
Flat    "     empty,       "        8     "  "      34    " 

Passenger  car,  empty,  weight  20  tons  )        t(      ^n    (( 

loaded,       "      25  "    \ 

Drawing  room  car,  "      35  "  3          "      50  to  60  feet. 

Sleeping-car,  weight,        30  to  45    "  "     50  to  70    " 

RESISTANCE    DUE   TO  INERTIA. 

14.  We  are  now  able  to  calculate  with  a  fair  amount  of  pre- 
cision the  Propelling  Force  of  an  engine  and  the  Total  Resist- 
ance opposed  to  it  at  any  given  speed.  The  Difference  between 
these  two,  such  as  is  represented  by  NM,  in  Diagram  II,  gives 
the  force  in  Ibs.  per  ton  which  goes  to  overcome  the  inertia 
of  the  train:  if  the  Propelling  Force  be  the  greater,  increasing 
the  velocity;  but  if  the  Resistance  be  the  greater,  decreasing  it. 

We  will  first  consider  the  subject  on  the  assumption  that  the 
accelerating  force  remains  constant  at  all  speeds,  and  that  there 
are  nofrictional  resistances. 

It  is  found  by  experiment  that  a  force  of  1  Ib.  acting  on  a 
weight  of  32.2  Ibs.  (which  is  perfectly  free  to  move  in  the  di- 
rection in  which  the  force  is  acting)  will,  after  acting  on  it  for  1 
second,  give  it  a  velocity  of  1  foot  per  second;  and  that  the 
velocity  at  all  points  increases  in  proportion  to  the  interval  of 


14  KAILROAD   LOCATION. 

time  during  which  the  force  acts:  also,  that  for  a  given  force, 
the  velocity  of  a  body  (after  it  has  been  acted  on  by  the  force 
for  a  certain  interval  of  time)  is  inversely  proportional  to  the 
weight  of  the  body.     Thus  the  value  of  the  Accelerating 
Force  in  Ibs.  per  ton  of  train  equals 
1.518V 
t     ' 

where  t  =  time  in  minutes  during  which  force  acts,  and  V  = 
velocity  in  miles  per  hour  acquired  in  time  t. 

But  this  formula  takes  no  account  of  the  force  necessary  to 
cause  the  wheels  to  rotate;  it  only  allows  for  motion  in  the  di- 
rection in  which  the  force  acts.  In  order  to  obtain  the  ad- 
ditional force  required  to  overcome  the  Rotative  Energy  of 
the  Wheels,  we  may  imagine  the  whole  weight  of  each  wheel 
concentrated  at  a  point  distant  from  its  axis  by  an  amount 
equal  to  the  Radius  of  Gyration  of  the  wheel.  For  ordinary 
rolling-stock  we  may  say  that  this  distance  equals  0.75  of  the 
radius  of  the  wheel;  and  the  velocity  with  which  a  point  so 
situated  rotates  round  the  axis  equals  0.75  the  velocity  of  the 
train.  Now  the  ratio  of  the  weight  of  the  wheels  to  the  total 
weight  of  a  train  of  medium  length  varies  from  about  0.1  to 
0.25,  according  to  whether  the  cars  are  loaded  or  empty,  the 
proportion  in  the  case  of  Passenger  Cars  being  about  the  same 
as  with  Loaded  Freight  Cars.  Therefore  the  Total  Force  neces- 
sary to  overcome  the  entire  Inertia  of  the  train  varies  from 
about 


to 


t 

where  F  =  constant  accelerating  force  in  Ibs.  per  ton  (2000 
Ibs.)  of  train. 

The  former  value  is  applicable  to  Loaded  and  the  latter  to 
Empty  cars. 

As  regards  the  distance  covered  by  the  train  from  the  start- 
ing-point to  the  point  at  which  it  attains  the  velocity  V,  it  can 
be  found  by  the  formula 


where  8  =  distance  in  feet. 

15.  Now  the  force  required  to  stop  a  train  travelling  with  a 
certain  velocity,  in  a  given  time,  equals  the  force  which  is 
necessary  to  give  it  that  velocity  in  the  same  time;  so  that  the 


RAILROAD   LOCATION".  15 

formula  given  above  for  F  applies  to  the  resistance  caused  by 
the  Application  of  Brakes,  as  well  as  to  the  Propelling  power 
of  the  engine.  Now,  since,— as  in  the  case  of  the  driving-  - 
wheels  of  a  locomotive,— as  soon  as  slipping  begins,  the  ad- 
hesion at  the  rails  decreases  rapidly,  therefore,  in  applying  the 
brakes,  the  pressure  should  be  such  that  the  wheels  will  just 
roll  on  the  rails;  i.e.,  the  resistance  on  the  brakes  must  not  be 
allowed  to  exceed  the  resistance  at  the  rails,  but  should  be  as 
near  to  this  limit  as  possible.  If  the  pressure  on  the  brakes 
could  be  adjusted  so  as  to  effect  this  in  practice,  we  should 
have  an  efficiency  for  the  brakes  equal  to  the  coefficient  of 
adhesion,  which  we  have  already  considered  under  ordinary 
circumstances  to  equal  0.2. 

But  it  is  found  that  with  Automatic  Brakes  we  cannot 
generally  rely  on  a  greater  efficiency  than  0.12,  which  is  equal 
to  a  value  of  F  (if  the  brakes  are  applied  to  the  whole  train) 
of  240  Ibs.  Thus  the  brakes  may  be  said  to  offer  a  resistance 
equivalent  to  a  12  p.  c.  grade. 

In  the  case  of  Hand  Brakes  it  usually  takes  about  four 
times  as  great  a  distance  in  which  to  stop  a  train  when  they 
are  used,  as  with  Automatic  ones  applied  to  the  whole  train. 

Suppose  under  the  above  assumption  we  have'  a  passenger- 
train  running  at  a  speed  of  60  miles  per  hour.  Ii  steam  is 
shut  off  at  the  same  instant  that  the  brakes  are  applied  auto- 
matically— with  an  efficiency  of  0.12 — to  three  quarters  of  the 
weight  of  the  train,  the  retarding  value  of  F  would  equal 
.75  X  240  =  180  Ibs.  per  ton,  and  thus  by  our  previous  formula 
gives  a  value  for  t  —  0.53  minutes,  from  which  we  can  obtain 
S  =  1400  feet.  Had  the  train  being  going  at  only  30  m.  p.  h. 
instead  of  60,  it  could  have  been  pulled  up  in  one  half  the 
time  and  one  quarter  the  distance  it  required  to  stop  it  when 
running  at  60  m.  p.  h.  Thus  in  order  to  stop  a  train  going 
at  60  m.  p.  h.,  we  must  apply  four,  times  the  amount  of 
brake-resistance  wliich  would  be  required  to  stop  it  if  going 
at  30  m.  p.  h.  in  the  same  time. 

16.  So  far  wre  have  dealt  only  with  a  change  of  velocity 
from  Rest  to  F,  or  from  F  to  Rest.  Suppose,  however,  in  the 
former  case  that  the  train,  instead  of  being  at  rest,  before  the 
accelerating  force  Fis  applied,  has  an  Initial  Velocity  (v). 
The  formulae  given  in  section  14  then  become  changed,  F  in 


16  RAILROAD   LOCATION. 


this  case  varying  from  about 


1.6(F-0)          1.7  (F-t>) 
"-  "   --  ' 


and 


And  just  as  the  previous  formulae  applied  to  either  an  accel- 
erating or  retarding  force,  so  these  apply  equally  well  to  the 
Propelling  Force  of  the  Locomotive  or  the  Resistance  of  the 
Brakes. 

As  an  Example,  suppose  we  take  a  Passenger-train  run- 
ning at  50  miles  per  hour.  The  value  of  F  necessary  to  re- 
duce this  speed  to  30  m.  p.  h.  in  one  minute  =  1.6  X  20  =  32 
Ibs.  per  ton,  which  gives  a  resistance  equivalent  to  a  -f-  1.6 
p.  c.  grade.  Problems  such  as  the  above,  where  the  value  of 
F  is  assumed  constant,  where  no  account  is  taken  of  the  fric- 
tional  resistances,  and  in  which  the  question  of  the  time  t  is 
not  directly  involved,  may  often  be  solved  more  simply  still 
by  means  of  the  Table  of  Equivalent  Heights  given  below. 

HEIGHT  CORRESPONDING  TO  VELOCITY. 

17.  In  the  above  example  of  the  train  running  at  60  m.  p. 
h.  being  brought  to  a  stand-still,  if  the  brakes  had  been  ap- 
plied to  the  whole  train  with  an  efficiency  of  240  Ibs.  per  ton, 
it  would  have  been  stopped  in  a  distance  of  about  1056  ft.  ;  or, 
putting  it  in  another  way,  the  train  could  have  run  up  a  12 
p.  c.  grade  fora  distance  of  1056  feet  before  stopping,  showing 
that  it  had  —  stored  up  in  it  —  the  Energy  necessary  to  raise  itself 
vertically  through  a  height  of  about  127  feet.  In  a  similar 
way  —  without  going  into  the  subjects  of  Kinetic  and  Potential 
Energy—  every  velocity  may  be  shown  to  have  a  corresponding 
vertical  height. 

Now  about  5.6  p.  c.  of  this  rise,  in  the  case  of  trains,  is  due  to 
the  Rotative  Energy  of  the  wheels  (when  dealing  with  loaded 
cars)  and  the  remainder  is  simply  the  height  from  which  a 
body  must  fall  under  the  influence  of  a  force  equal  to  its  own 
weight,  —  i.e.  .gravity,  —  in  order  to  obtain  the  velocity  in  ques- 
tion. But  since  this  Rotative  Energy  is  taken  account  of  in  the 
previous  formulae,  we  can,  by  finding  the  value  of  5  when  F  — 
2000,  obtain  for  any  given  velocity  the  corresponding  vertical 
height. 


KAILROAD  LOCATION. 


17 


In  this  way  the  following  table  has  been  calculated  for  Pas- 
senger or  Loaded  Freight  Cars. 

For  a  train  of  Empty  Freight  or  Flat  Cars,  6  p.  c.  should  be 
added  to  the  heights  given. 

TABLE  OF  HEIGHTS  IN  FEET  CORRESPONDING 
TO  VELOCITY  IN  MILES  PER  HOUR. 


Vel. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

3.5 

4.3 

5.1 

5.9 

6.9 

7.9 

9.0 

10.2 

11.4 

12.7 

20 

14.1 

15.5 

17.0 

18.6 

20.2 

22.0 

23.8 

25.7 

27.6 

29.6 

30 

31.7 

33.8 

36.0   38.3 

40.7 

43.1 

45.6 

48.2 

50.8 

53.5 

40 

56.3 

59.2 

62.1 

65.1 

68.2 

71.3 

74.5 

77.8 

81.2 

84.6 

50 

88.0 

91.5 

95.1 

98.9 

102.7 

106.5  110.4  114.4 

118,4 

122.5 

60 

126.7 

131.0 

135.3  139.7 

144.2 

148.7  153.3  '158.0 

162.8 

167.6 

70 

1172.5 

177.4 

182.5 

187.6 

192.8 

198.0  ,203.3  |208.7 

214.2 

219.7 

Now  if  we  have  a  Passenger  train  running  at  a  speed  of  20 
m.  p.  h.,  and  we  wish  to  know  what  its  velocity  will  be  after 
descending  1000  feet  of  a  3  p.  c.  grade — ignoring  as  before 
frictional  resistances — we  can  find  it  at  once  from  the  Table, 
thus :  Its  velocity  at  the  foot  of  the  grade  will  be  that  due  to  the 
height  corresponding  to  a  velocity  of  20  m.  p.  h.  -}-  30  feet  = 
44.1  feet,  which  corresponds  with  the  velocity  required,  name- 
ly, 35.4  miles  per  hour.  Or,  suppose  we  wish  to  know  what 
rate  of  grade  would  be  required  to  decrease  the  speed  of  the 
above  train  from  40  rn.  p.  h.  to  25  m.  p.  h.  in  a  distance  of  1000 
ft.:  we  have 

Height  corresponding  to  40  m.  p.  h.  =  S6.-3  feet 
"25        "       =22.0'    " 

Difference  =  34.3  feet. 

Thus  it  is  a  3.43  p.  c.  grade  that  would  be  required. 

18.  So  far  we  have  dealt  only  with  the  Inertia  of  the  train  on 
the  supposition  that  the  propelling  force  of  the  engine  is  con- 
stant at  all  speeds,  and  that  there  are  no  frictional  resistances. 
A  method  much  in  use  in  practice  which  partially  corrects  for 
both  these  fallacies  is  that  of  allowing  for  the  mean  frictional 
resistance  and  the  mean  propelling  force  of  the  engine,  and  then, 
by  the  aid  of  formulae  similar  in  effect  to  those  given  above, 
obtaining  approximate  values  of  8. 

19.  But  this  method  of  averaging  gives  very  unreliable  re- 
sults when  dealing  with  any  but  comparatively  low  velocities, 
so  that  the  following  Graphic  Method,,  which  is  extremely 


18  KAILROAD   LOCATION. 

simple,  is  in  most  cases  preferable,  since  the  correctness  of 
the  results  obtained  by  it  depends  almost  solely  on  the  care 
employed  in  working  it. 

Let  the  Lines  of  Resistance  and  Propelling  Force  be  drawn 
as  in  Diagram  II. 

00 

Take  anyordiuate  NQ,  and  make  PQ  =          * 


Similarly  take  other  ordinates,  and  thus  fix  other  positions 
of  the  point  P. 

Draw  the  curve  OPD  through  these  points.  Then,  if  (as  in 
Diag.  II)  1  inch  vertical  =10  Ibs.,  and  1  inch  horizontal  =  20 
miles  per  hour,  the  area  (shown  shaded  in  Diag.  II)  enclosed 
by  the  curve  OPD,  the  line  OS,  and  the  ordinate  corre- 
sponding to  any  given  velocity  gives  the  distance  covered 
while  attaining  that  velocity,  using  as  a  scale  1  square  inch 
=  1  linear  mile.  (See  Note  B,  Appendix.)  And  as  a  conse- 
quence of  this,  assuming,  e.g.,  the  train  has  an  Initial  velocity 
of  20  miles  per  hour,  and  a  final  velocity  of  34  miles  per  hour, 
the  area  between  the  ordiuates  of  20  and  34  in.  p.  h.  gives  the 
distance  traversed  while  the  speed  is  being  raised  from  the 
lower  velocity  to  the  higher, 

By  the  ordinary  method  of  averaging,  at  a  speed  of  34  m. 
p.  h.  the  distance  would  be  represented  by  the  area  Opg,  in- 
stead of  the  shaded  portion.  This  shows  the  little  dependence 
to  be  placed  on  the  averaging  process,  when  dealing  with 
speeds  which  approach  the  limit. 

But  there  is  a  correction  to  apply  to  this  if  we  wish  to 
allow  for  the  Rotative  Energy  of  the  wheels  ;  and  this,  as  we 
have  already  seen,  varies  from  about  6  to  12  p.  c.  of  the  total 
energy  of  the  train  ;  so  that  in  the  case  of  Passenger  or 
Loaded  Cars  6  p.  c.  should  be  added  to  the  distance  as  ob- 
tained above,  and  in  the  case  of  Empty  Cars  12  p.  c. 

20.  This  method  may  be  applied  to  a  variety  of  problems  i-u 
Railroad  Dynamics  :  thus,  for  example,  suppose  we  have  a 
train  travelling  at  60  m.  p.  h.,  and  we  wish  to  know  how  far 
it  will  run  if  the  brakes  are  suddenly  applied,  causing  an  ad- 
ditional resistance  of  20  Ibs.  per  ton  —  of  entire  train.  Then 
the  line  of  total  resistance  will  be  given  by  the  dotted  line 
EG  (Diag.  II),  and  the  value  of  MN  at  any  given  speed  will 
equal  the  entire  ordiuate  from  OH  to  the  curve  EG,  for  the 

*  AH  measured  in  inches  on  the  diagram, 


RAILROAD   LOCATION. 


19 


line  of  propelling  force  then  coincides  with  OH— i.e.,  equals 
zero.  Or,  conversely,  if  the  train  be  pulled  up  in  any  known 
distance,  we  can  by  two  or  three  trials  ascertain  the  efficiency 
of  the  brakes.  If  in  dealing  with  such  problems  as  these  we 
have  in  the  course  of  the  distance  travelled  various  rates  of 
grade  and  curves  of  different  "  degree,"  we  can,  without 
serious  error,  draw  our  line  of  resistance  for  the  mean  grade 
and  the  mean  degree  of  curvature. 

21.  We  are  now  able  to  ascertain  the  effects  of  various 
amounts  of  Rise  and  Fall  on  the  velocity  of  a  train.  In  the 
first  place,  we  will  go  back  to  our  former  assumption  that  the 
engine  exerts  the  same  tractive  force  at  all  speeds,  and  that 
there  are  practically  no  frictional  resistances.  Of  course  this 
is  a  thoroughly  erroneous  supposition,  but  by  adopting  it  we 
simplify  matters  very  considerably,  and  yet  at  the  same  time 
are  able  to  obtain  results  which,  for  practical  purposes,  are 
sufficiently  correct  when  we  limit  their  application  to  compar- 
atively short  distances. 


FIG.  8. 

In  Fig.  2  let  ABGDEF  represent  the  grades  on  a  lim- 
ited portion  of  a  certain  road,  then — under  the  assumption  al- 
ready made — if  we  have  a  train  running  along  the  level  tow- 
ards A  at  a  uniform  speed  of  40  miles  per  hour,  we  obtain 
from  the  Table  of  Equivalent  Heights  in  Sec.  17— 
Vel.  Head  in  ft.  at  A  =  56,  because  V  =  40  m.  p.  h. 

"        #  =  56-40  =  16;   .'.  Fat  5  =  21m.p.h. 

£=16  +  10  =  26;   .-.     "     (7  =  27      " 
"       £  =  264-30  =  56;   .-.     "    Z>  =  40 

j£=  56  +  30  =  86;   .'.     "    #=50      " 
"  "        ^=60-30  =  56;  .'.     "    F=±Q      " 


20  RAILROAD   LOCATION. 

By  determining  the  speed  at  a  few  such  points  as  these,  and 
drawing  through  them  the  dotted  lines  as  in  Fig.  2,  we  have 
practically  a  Profile  of  Velocities,  from  which  we  can  read 
approximately  the  speeds  at  different  points  on  the  grade. 

22.  In  such  a  case  as  the  above  the  strain  on  the  draw-bar 
of  the  engine  would  at  all  points  be  constant,  and  the  amount 
of  work  done  in  transporting  the  train  from  A  to  F  would— 
ignoring  the  difference  in  distance,  which  of  course  in  prac- 
tice amounts  to  nothing — be  the  same  whether  the  train  went 
along  the  grade  ABEF,  or  along  a  level  grade  ADF. 

Now  the  effect  of  running  over  such  a  ridge  as  ABD  is  to 
lower  the  average  speed:  thus  if  running  from  A  to  D  on  the 
level,  the  train  would  arrive  at  D  much  sooner  than  by  way 
of  ABD.  Again,  in  running  over  the  grade  DEF,  its  average 
velocity  would  be  much  higher  than  along  the  level  DE.  Thus 
the  ridge  ABD  is  detrimental  to  high  speeds,  but  the  depres- 
sion  DEF  tends  to  raise  the  average  velocity.  In  dealing  with 
cases  where  the  distance  AD  or  DF  does  not  exceed  a  few 
hundred  yards,  the  results  obtained  as  above  are  sufficiently 
accurate  to  enable  the  engineer  to  find  the  effect  of  adopting 
certain  grades  over  such  a  ridge  as  D  or  depression  E. 

23.  But  this  theory  utterly  fails  when  applied  to  grades  of 
considerable  length,  for  the  reason  that  the  possible  tractive 
power  of  the  engine— at  any  but  the  lower  speeds — decreases 
as  the  velocity  increases,  and  the  resistances  increase  rapidly  as 
the  speed  is  raised. 

We  will  now  consider  the  result  of  taking  these  considerations 
into  account  in  the  case  shown  in  Fig.  2.  Now  if  the  train 
comes  on  to  the  grade  AB  at  a  certain  speed — assuming  that 
the  Effective  Horse-power  remains  constant — it  will  have  a  ve- 
locitv  at  B  appreciably  greater  than  that  which  we  should  ob- 
tain for  it  at  that  point  by  means  of  the  Table  of  Equivalent 
Heights.  So  also  at  D  it  will  have  a  velocity  greater  than  it 
had  at  A,  although  by  the  Table  the  velocity  at  A  and  D  should 
be  the  same.  The  reason  of  this  is,  that  the  increase  in  the 
accelerating  force  is  more  than  in  proportion  to  the  increase 
in  the  total  propelling  force,  being  due  to  a  decrease  in  the  re- 
sistances as  well  as  to  the  reduction  in  speed.  Similar  reasoning 
applies  to  the  down-grades  BC  and  CD,  so  that  by  the  time  the 
.train  has  got  to  D  the  total  amount  of  work  done  on  the  higher 
grade  is  relatively  less  than  what  it  would  have  been  along 


RAILROAD  LOCATION.  21 

the  level  AD,  owing  to  the  reduced  frictional  resistances.  Thus 
the  train  is  travelling  faster  at  D  than  it  was  at  A,  although  it 
has  lost  time  on  the  way.  Similarly,  in  the  case  of  crossing  a 
depression  such  as  E,  the  amount  of  work  done  will  be  greater 
by  the  lower  route  than  along  the  level,  and  the  train  will  thus 
have  at  F  a  velocity  less  than  it  had  at  D,  although  it  will  have 
made  better  time  between  D  and  F  by  way  of  E,  than  along 
the  level  DF. 

But  although  the  train  arrives  at  D  with  a  higher  velocity 
than  if  it  had  proceeded  along  the  level,  yet  this  increase  in 
velocity  only  partially  makes  up  for  the  time  lost  between 
A  and  D.  So  also  the  decrease  in  speed  at  F  does  not  entirely 
counteract  the  gain  in  time  made  along  DEF. 

The  amounts  by  which  the  velocities  at  D  and  F  actually 
differ  from  those  obtained  by  the  Table,  depends  mainly  in 
practice  on  the  distance  between  A  and  D,  or  D  and  F.  The 
greater  these  distances  are,  the  less  reliance  is  to  be  placed  on 
the  Table  ;  so  much  so  in  fact  in  dealing  with  long  grades,  as 
to  render  the  energy  of  the  train  itself — considered  as  a  store 
of  available  tractive  power — practically  worthless. 

24.  It  is  usual  for  Railroad  Companies  to  adopt  a  cer.tain 
rate  of  grade  which  is  not— except  where  Pusher-grades 
are  used — to  be  exceeded.  This  is  usually  termed  the  Maxi- 
mum or  Ruling  Grade,  and  is  selected  with  due  considera- 
tion to  the  tractive  power  of  the  locomotives  to  be  employed, 
the  probable  amount  of  traffic,  the  weight  of  trains  to  be 
hauled,  and  the  speed  required  to  be  maintained.  It  is  also 
selected  in  most  cases  so  as  to  admit  of  a  train  starting  on  the 
grade,  if  by  any  chance  it  should  have  had  to  pull  up.  Also, 
it  should  be  such  that  the  locomotive  employed  can  haul  the 
train  over  it,  altogether  independent  of  the  Momentum — or 
more  correctly  Energy — of  the  train.  By  means  of  Diagram 
II  we  can  readily  select  the  most  suitable  Maximum  Grade 
by  drawing  the  line  of  resistance — for  a  level  track — and  the 
line  of  propelling  force  suitable  for  the  locomotives  to  be  em- 
ployed ;  the  length  of  the  ordinate  NM,  when  scaled  off, 
gives  the  equivalent  resistance  in  Ibs.  per  ton  of  the  maxi- 
mum grade.  Thus,  in  the  case  of  the  example  given  in 
Diagram  II,  if  the  speed  required  to  be  maintained  on  the 
grade  equals  24  miles  per  hour, — since  NM  represents  to 
scale  about  17  Ibs.  per  ton, — the  maximum  grade  will  equal 


22  RAILROAD  LOCATION", 

0.85  p.-c.  Had  the  required  speed  been  only  10  miles  per 
hour,  we  might  then  have  used  a  1.6  p.  c.  grade.  But  proba- 
bly in  neither  of  these  cases  could  the  train  start  on  the  grade, 
and  in  order  to  allow  for  this,  we  must  assume  that  the  line 
of  resistance  at  no  point  dips  below  15  Ibs.  per  ton,—  i.e.,  12 
Ibs.,  in  accordance  with  Sec.  3,  and  a  small  margin  of  3  Ibs. 
to  overcome  the  Inertia  of  the  train. — Thus,  allowing  for  stop- 
pages, if  a  speed  of  24  m.  p.  h.  is  to  be  maintained  in  the  case 
shown  in  Diagram  II,  the  maximum  grade  must  not  exceed 
0.55  p.  c. ;  but  if  10  m.  p.  h.  only  is  required,  then — includ- 
ing allowance  for  stoppage — the  maximum  grade  may  be  1.1 
p.  c.  But  we  must  remember  that  where  the  velocity  required 
to  be  maintained  on  the  maximum  grade  exceeds  that  given 
by  Aa,  in  Sec.  10,  some  allowance  should  be  made  for  the 
probable  increase  in  boiler-pressure  after  the  train  has  come 
to  a  stand-still ;  which  means  that  on  starting,  the  I.  H.  P.  of 
the  engine  may  be  placed  considerably  above  its  normal 
working  power. 

25.  Without  going  into  the  question  of  the  Economy  of  the 
Steam  engine,  we  may  say  that  a  Locomotive  works  with  its 
greatest  efficiency  when  the  boiler-pressure  remains  constant 
and  the  engine  is  running  at  a  uniform  velocity.  Thus  fluc- 
tuations in  speed  or  variations  in  the  opposing  resistances  are 
more  or  less  detrimental  to  the  working  of  the  locomotive. 

As  a  consequence  of  this,  if  a  certain  elevation  has  to  be  at- 
tained, in  order  to  make  the  work  as  easy  on  the  engine  as 
possible,  the  grade  should  be  such  as  to  render  the  sum  of  the 
resistances  opposed  at  all  points  as  nearly  constant  as  possible. 
Thus,  if  the  alignment  be  straight,  the  rate  of  grade  should 
be  uniform  ;  but  if  curves  or  other  irregularities  occur,  they 
should  be  compensated  for,  so  that  a  constant  resistance  may 
be  maintained. 

2C.  Compensation  for  Curvature. — From  Diagram  I  we 
see  that  at  10  miles  per  hour  the  resistance  for  each  degree  of 
curvature  is  about  1  Ib.  per  ton,  i.e.,  equivalent  to  a-|-  0.5  p.  c. 
grade,  and  that  at  about  30  in.  p.  h.  it  is  about  half  this.  The 
rate,  however,  usually  adopted  is  .03  p.  c.,  which  is  suitable  to 
a  speed  of  about  25  m.  p.  h.  Thus,  if  the  equivalent  grade  on 
a  tangent  is  1.5  p.  c.,  we  must  reduce  it  on  a  3°  curve  to  1.41 
p.  c.  in  order  that  the  resistance  may  remain  constant. 


RAILROAD  LOCATIOK.  23 

27.  Compensation  for  Brakes,  etc.— A  point  to  be  re- 
membered in  running  a  long  uniform  grade  which  does  not 
approach  the  maximum  is  to  consider  at  what  points  the  train 
will  be  required  to  slacken  or  increase  its  speed.     For  exam- 
ple, suppose  on  such  a  grade  we  have  a  sharp  curve  around 
which  the  speed  is  not  to  exceed  20  miles  per  hour,  but  that 
on  the  tangent  at  either  end  of  it  a  speed  of  40  m.  p.  h.  can  be 
maintained.     By  means  of  the  Table  of  Equivalent  Heights  we 
can  adapt  the  Energy  of  the  train  so  that  the  velocity  will  be 
reduced  without  the  application  of  the  brakes,  and  that  when 
the  curve  is  passed  the  speed  of  the  train  can  be  more  readily 
increased  from  20  to  40  in.  p.  h.    But  in  doing  this  we  have 
to  be  careful  that  at  the  lower  end  of  the  curve  we  do  not  in- 
crease the  grade  so  as  to  tax  the  engine  too  severely.     At  all 
such  points  as  crossings,  where  short  stoppages  are  required, 
attention  should  be  paid  to  this,  for  by  so  doing  we  can  at 
times  save  something  even  in  the  cost  of  construction,  besides 
saving  considerably  in  fuel  and  in  wear  and  tear  to  the  Roll- 
ing-stock. 

28.  But  though  the  operating-expenses  may  be  reduced  to 
a  minimum  by  the  use  of  Long  uniform  (equivalent)  grades, 
the  amount  necessarily  expended  on  their  construction  may 
be  too  great  to  warrant  adopting  them.  In  such  cases  Broken 
Grades  have  then  to  be  used. 

Now  we  have  already  seen  how  to  obtain  the  effect  of  un- 
dulations on  the  velocity  and  the  work  done,  so  that  we  can 
in  any  particular  case  determine  for  ourselves  what  will  be 
the  result  of  selecting  a  certain  arrangement  of  grades.  The 
following  "pointers,"  however,  deduced  from  what  has  al- 
ready been  said,  may  come  in  handy. 

1.  A  Rise  from  the  uniform  grade  is  detrimental  to  fast 
traffic,  and  though  there  is  a  saving  in  actual  work  done  on  it, 
there  is  probably  no  saving  in  the  consumption  of  fuel. 

2.  A  Depression  from  the  uniform  grade  tends  to  increase 
the  mean  velocity,  but  at  the  cost  of  a  considerable  amount  of 
extra  fuel. 

3.  Breaks  in  the  grade  which — from  the  point  where  the 
broken  grade  leaves  the  uniform  one  to  the  point  where  they 
next  intersect — do  not  exceed,  say,  1000  to  2000  feet,  may  be 
regarded  as  "Momentum  Grades,"  and  accordingly  are  not  so 
injurious  as  longer  breaks  where  the  Initial  Energy  of  the 


24  EAILROAD  LOCATION". 

train  is  small  compared  with  the  Total  Energy  to  be  expended 
on  them. 

4.  The  nearer  the  uniform  grade  approaches  the  "Maxi- 
mum grade,"  the  more  injurious  do  any  breaks  become;  and 
the  only  point  in  connection  with  the  "Maximum  grade," 
where  an  increase  in  the  rate  is  allowable,  is  the  insertion  of  a 
"Momentum  grade"  at  its  lower  end. 

5.  Breaks  in  a  grade  are  more  injurious  to  slow  than  to  fast 
traffic— as  may  be  seen  from  the  Table  of  Equivalent  Heights 
— e.g.,  an  increase  in  elevation  of  20  feet  reduces  the  velocity 
from  30  to  Ifrmiles  per  hour,  while  a  velocity  of  60  m.  p.  h. 
is  only  reduced  to  about  55  miles  per  hour. 

6.  Be  careful  in  inserting  Momentum  grades  that  they  will 
not  be  such  as  to  cause  the  velocity  at  any  point  to  exceed  the 
safe  limit.     A  difference  in  elevation  of  about  30  feet  be- 
tween the  Broken  and  the  Uniform  grade  should  generally  be 
taken  as  a  limit. 

29.  Another  point  to  be  considered,  which  we  have  not  yet 
referred  to,  is  the  increase  in  Liability  to  Danger  of  Break- 
ing-train and  Derailment  to  which  an  undulating  grade 
gives  rise.    For,  suppose  in  Fig.  2  we  have  a  train  running  up 
the  grade  from  A  to  B:  as  soon  as  the  engine  is  over  the 
summit  the  pull  on  the  draw-bar  becomes  enormously  in- 
creased, and  similarly  with  the  car-couplings  throughout  the 
entire  train;  so  that,  unless  the  greatest  care  is  taken  in  ap- 
plying the  brakes,  the  train  runs  a  very  great  risk  of  being 
broken  in  two.     Similarly,  in  such  a  hollow  as  E,  the  cars 
near  the  centre  of  the  train  are  liable  to  get  terribly  jammed 
together,  thereby  greatly  increasing  the   chances  of  Derail- 
ment. 

Vertical  curves  reduce  these  dangers  considerably,  but  not 
entirely. 

It  must  be  remembered  that  it  is  not  in  the  least  necessary 
that  one  of  the  grades  should  be  an  up-grade  and  the  other  a 
down-grade:  it  is  the  difference  in  flie  rate  of  grade  that  has  to 
be  looked  out  for.  (See  Sec.  100.) 

30.  In  Fig.  3,  let  ACB  and  ADB  represent  two  different 
routes  between  A  and  B,  the  total  Rise  and  Fall  between  the 
two  points  in  each  case  being  the  same.     The  amount  of  work 
done  in  hauling  the  train  from  A  to  B  by  way  of  G  will, 
supposing  we  are  dealing  with  grades  so  long  that  the  ques- 


RAILROAD   LOCATION.  25 

tion  of  "Momentum  Grades"  may  be  ignored,  be  then  prac- 
tically the  same  as  by  way  of  D.  Similarly,  if  such  a  point  as 
H  in  Fig.  4  has  to  be  reached,  the  work  done  in  hauling  the 
train  along  the  uniform  grade  EH  will  be  practically  the  same 
as  by  way  of  FG.  It  is  not  the  amount  of  work  done  on  the 
grades  themselves  that  has  to  be  considered,  but  the  amount  of 
extra  work  which  is  uselessly  done  by  a  heavy  engine  haul- 
ing a  large  surplus  of  dead-weight  (due  to  its  own  size)  over 


FIG.  3. 


grades  where  a  lighter  engine  could  have  hauled  the  train 
equally  well.  If  each  of  the  divisions  EF,  FG,  and  GRwere 
a  suitable  length  for  one  engine  to  work,  the  lower  route 
would  then  be  as  economical  probably  as  regards  Operating 
Expenses  as  the  higher.  Besides  this,  we  have  the  increased 


Fia.  4. 

consumption  of  fuel,  before  referred  to,  which  always  accom- 
panies variations  in  grade. 

If  we  make  each  of  the  divisions  along  the  lower  route  from 
E  to  //of  such  a  length  as  to  keep  the  engine  employed  on 
each  fairly  busy, — using  a  different  engine  on  each  division, — 
the  lower  route  is  then  as  economical  as  can  be  wished  for, 
but  otherwise  the  upper  route  has  the  advantage. 


26  RAILROAD  LOCATION. 

31.  Now  the  average  length  of  an  Engine-stage  may  be 
considered  to  be  about  100  miles,  which  is  of  course  too  long 
to  enable  us  to  work  the  lower  route  in  the  manner  described 
above.  We  may  often,  however,  by  adopting  a  Pusher-grade, 
even  at  a  point  where  at  first  it  appears  unnecessary,  make 
a  decided  improvement  in  the  economy  of  our  grades.  The 
length  of  this  grade,  if  the  Pusher  is  to  be  kept  steadily  em- 
ployed, depends  of  course  on  the  number  of  trains  to  be  taken 
up  it  each  day:  if  there  are  four  trains  a  day  the  engine  will  be 
kept  sufficiently  at  work  if  the  length  of  the  grade  is  only  12 
miles.  As  to  the  rate  of  grade  which  may  be  adopted  in  such 
cases  as  this,  Mr.  Wellington  gives  the  following  Table,  which 
is  suitable  for  average  Consolidation  Engines,  the  coefficient 
of  adhesion  being  taken  at  0.25  : 


TABLE  OF  PUSHER-GRADES. 


Grade  worked  by 

Net  Load  of 

GRADE  POSSIBLE  WITH— 

1  Pusher. 

2  Pushers. 

Level. 

2675 

0.38 

0.74 

0.2 

1758 

0.75 

1.26 

05 

1147 

1.30 

2.01 

1.0 

711 

2.16 

3.13 

1.5 

504 

2.96 

4.13 

2.0 

383 

3.72 

5.03 

32.  Maximum  Curvature. — In  countries  where  construc- 
tion is  comparatively  easy,  it  is  often  the  custom  to  select  a  cer- 
tain degree  of  curvature  which  is  not  to  be  exceeded.  The  ques- 
tion of  the  speed  required  to  be  maintained  is  the  main  one 
which  arises  in  this  case.  Wear  and  tear  of  rails  and  rolling- 
stock  is  also  an  important  factor.  The  question  of  resistance 
— at  ordinary  speeds — is  comparatively  unimportant,  since  at 
a  speed  of  25  miles  per  hour  a  10°  curve  only  offers  the  resist- 
ance of  about  a  0.3  p.  c.  grade.  In  rough  country  it  is  im- 
possible to  fix  a  "maximum," for  the  additional  cost  of  con- 
struction which  the  adoption  of  a  limiting-grade  might  involve 
would  perhaps  be  an  inconceivably  greater  consideration  than 
the  loss  of  a  few  seconds— or  possibly  minutes — in  time.  As 
regards  the  question  of  the  Safe  Speed  on  Curves,  it  is  diffi- 


RAILROAD   LOCATION".  2 

Cult  to  lay  down  any  law,  but  it  is  supposed  to  vary  inversely 
as  the  square  root  of  the  radius.  Thus  if  we  assume  that  40 
miles  per  hour  is  a  safe  speed  on  a  2°  curve,  the  speed  should 
be  limited  to  20  m.  p.  h.  on  an  8°  curve  and  to  14  m.  p.  h.  on 
a  16°  curve.  The  chances  of  derailment  and  the  wear  and 
tear  of  rolling-stock  and  rails  are  decreased  materially  by  the 
use  of  Transition  curves.  (See  Sec.  96  ) 

33.  It  is  almost  unnecessary  to  refer  to  the  subject  of  Re- 
verse Curves.   In  Station-yards,  where  the  speeds  are  insignifi- 
cant, their  use  is  sometimes  advisable;  but  on  the  Main  Track 
an  intervening  tangent  of  at  least  200  feet  in  length  should  be 
regarded  as  an  absolute  necessity.     A  fault  much  more  fre- 
quently found  is  the  insertion  of  a  short  tangent  between 
two  curves  of  the  same  direction.     Getting  on  to  a  tangent 
from  a  curve  is  as  hard  work  as  getting  on  to  a  curve  from  a 
tangent;  and  since  it  is  at  the  P.  C.  and  P.  T.  that  the  curve 
gives  its  maximum  resistance,  the  curves  should  at  least  be 
compounded  so  as  to  make  the  radius  of  curvature  at  all  points 
as  uniform  as  possible,  for  in  each  case  the  total  amount  of 
curvature  will  be  the  same.     Another  point   to  be  remem- 
bered— though  it  is  not  often  that  it  can  be  applied — is,  that  a 
road  which  has  its  curves  at  points  where  the  speed  is  com- 
paratively low  has  a  decided  advantage  over  one  in  which  the 
curves  are  located  at  places  where  a  high  speed  is  required  to 
be  maintained.     Thus,  if  a  certain  amount  of  curvature  has  to 
be  got  in,  in  such  a  place  as  DEF  in  Fig.  2,  it  should  be 
arranged  if  possible  so  that  the  curvature  at-Z)  and  F  will  be 
sharper  than  at  E.    Curvature  should  also  be  avoided  as  much 
as  possible  at  all  points  where  a  stoppage  is  required,  for  on 
starting,  the  resistance  due  to  the  curvature  is  a  great  con- 
sideration,  and,    as  we  saw  in  Sec.  6  and  Diagram  I,  will 
probably  make  it  as  difficult  for  the  train  to  start  as  a  decided 
up-grade. 

34.  We  have  now  dealt  in  a  more  or  less  superficial  way 
with  most  of  the  mechanical  problems  which  arise  in  connec- 
tion with  railroad  trains;  but  it  is  convenient,  for  the  sake  of 
more  readily  comparing  the  value  of  the  various  resistances 
to  passenger  and  freight  trains  at  average  speeds,  to  tabulate 
their  mean  values  (as  given  by  Prof.  Jameson)  as  follows: 


28 


BAILROAD   LOCATION. 


TABLE  SHOWING  COMPARATIVE  VALUES  OF  RESISTANCES 
AS  REGARDS  WORK  DONE. 


Items. 

Distance. 

Curvature. 

Rise  and  Fall. 

1  mile 

5280     feet 

600° 

25  0     feet 

1°  Curvature    

88" 

0  041     " 

1  foot  Rise  and  Fall- 

211.2    " 

24° 

1.0 

"  Rise  and  Fall  "  of  course  means  in  one  direction  only,  and 
is  so  stated  in  order  to  take  account  of  the  Rise  when  run- 
ning in  the  opposite  direction.  Thus  in  Fig.  3  the  total  Rise 
and  Fall  between  A  and  B  by  either  route  equals  710  feet. 

COST  OF  OPEEATING. 

35.  The  expense  involved  in  overcoming  the  resistances  re- 
ferred to  in  Sec.  34  is  not  proportional  to  the  amount  of  work 
which  is  performed  on  account  of  them.  For  instance,  it  is 
found  by  experience  that  hauling  a  train  over  one  mile  of  level 
track  costs  on  an  average  about  the  same  as  150  feet  of  rise 
and  fall, — not  of  25  feet,  as  given  in  the  last  table.  Similarly, 
with  curvature,  the  operating  of  one  mile  of  level  track  is 
found  to  cost  the  same  as  about  900°  of  curvature  (not  600°); 
so  that  as  regards  operating -expenses  the  table  given  in  Sec.  34 
becomes — 


Items. 

Distance. 

Curvature. 

Rise  and  Fall. 

1  mile 

5280      feet 

900° 

150        feet 

1°  Curvature  
1  foot  Rise  and  Fall. 

5.86    " 
35.2      " 

1° 
6° 

0.166    " 
1.0      .  " 

As  soon,  then,  as  we  know  the  expense  of  operating  one  mile 
of  level  track,  we  can  by  means  of  this  table  find  the  probable 
cost  of  working  any  certain  grade  or  any  given  amount  of 
curvature. 

36.  Taking  $1.00 — it  is  probably  nearer  90  cts.— as  the  aver- 
age cost  of  operating  one  mile  of  level  track  on  American 
Railroads  for  each  train  that  runs  over  it  (and  returns)  each 
day,  we  can  make  this  our  unit  of  operating-expenses  and 


RAILROAD   LOCATION. 


29 


term  it  the  cost  of  one  Train-mile.    The  items  which  go  to 
make  up  the  expense  of  the  train-mile  are  as  follow: 

(  Oil,  Fuel,  Waste. 

Motive  Power  .....  -j  Driver,  Fireman. 
(  Repairs. 

..  j  ^ir^ndBenewals  to  Cars. 


Train  Expenses. 
Road  Repairs 


Track,  Road-bed,  Structures. 


roT1      •,  (  Stations,  Terminal,  Taxes. 

'  '  ]  Repairs  and  Renewals. 

Taking,  then,  $1.00  as  the  cost  per  train-mile,  and  assuming 
the  interest  on  the  amount  capitalized  at  6  p.  c.,  we  obtain  the 
following  table: 


Unit. 

Value  per  annum  per 
daily  train. 

Amount  Capitalized. 

1  mile  

$350 

$5  833  33 

1  foot 

0  066 

1  10 

1°  Curvature  
1  foot  Rise  and  Fall..  .  . 

0.39 
2.33 

6,50 
38.88 

This  assumes  that  each  "  daily"  train  only  runs  350  days  in 
the  year,  which  makes  a  sort  of  allowance  for  Sundays, 
1 '  specials, "  etc. 

37,  From  the  above  we  see  that  if  we  have  ten  trains  mak- 
ing the  round-trip  every  day,  we  are  entitled  to  spend  $58,333 
extra  on  the  construction  of  a  certain  route,  if  by  so  doing  we 
can  save  a  mile  of  level  track;  so  also  we  should  be  entitled  to 
spend  $388  in  the  reduction  of  a  foot  of  rise  and  fall.  Thus 
with  10  daily  trains  we  might  safely  expend  2  X  $388  = 
$776  in  lowering  (only  one  foot)  such  a  summit  as  C  in  Fig. 
3;  but  if  C  had  been  the  terminus  of  the  line  AC  we  ought 
only  to  spend  $388  in  lowering  it  one  foot. 

Suppose  again  we  have  two  routes  to  select  from,  one  of 
which  would  probably  cost  $40,000  more  than  the  other,  but 
would  shorten  the  distance  by  one  mile  and  would  save  a  rise 
and  fall  of  100  feet.  Then  if  there  are  only  likely  to  be  three 
trains  running — including  returning — each  day,  we  are  not 
entitled  to  spend  more  than  ($5833+  $3888)  X  3  =  $29,163 
to  save  the  above  distance  and  rise  and  fall ;  therefore  it  would 
probably  be  injudicious  to  adopt  the  more  expensive  route. 


30  RAILROAD    LOCATION. 

38.  As  regards  the  cost  of  operating  Pusher  grades, 

we  find  that  a  Pusher  kept  pretty  busy  costs  on  aii  average 
about  $280  per  mile  of  incline  per  annum — i.e.,  $140  per 
mile  run — "  all  that  the  engine  fails  to  do  below  100  miles  per 
day  may  be  assumed  to  cost  f  roin  £  to  £  as  much  as  if  it  had 
been  rim,  and  is  so  much  added  to  the  cost  of  what  is  run." 
Thus  on  a  5-mile  incline,  with  only  4  trains  to  be  taken  up  it 
each  day,  the  probable  annual  expense  of  the  Pusher  will  be 
found  thus: 

Work  done,     4  X  5  X  $280  =  $5,600 
Work  not  done,  30  X  —^  =    2,100 

Total $7,700 

Had  we  been  able  to  reach  the  summit  without  adopting  a 
Pusher-grade — supposing  the  total  rise  and  fall  to  be  1000  feet 
— the  cost  of  "Rise  and  Fall "  would  have  been  for  the  4 
daily  trains  4  X  1000  X  $2.33  =  $9320,  representing  a  differ- 
ence in  the  operating-expenses  of  $1620  per  annum,  which 
at  6  p.  c.  would  have  warranted  our  expending  $27,000  more 
on  the  route  which  involved  the  Pusher-grade,  assuming 
curvature  and  distance  to  be  the  same  in  both  cases. 

39.  To  test  the  merits  of  different  routes  as  regards  operat- 
ing-expenses, we  may  express  them  in  terms  of  their  Equiv- 
alent Lengths  (L)  in  miles  thus: 

L  =  l  +  m+m' 

where 

I  =  actual  length  in  miles, 
H  =  total  rise  and  fall  in  feet, 
C  =  total  curvature  in  degrees. 

40.  As  regards  the  increase  in  operating-expenses  caused  by 
any  slight  increase  in  distance,  such  as  is  the  result  of  changes 
in  the  alignment,  it  is  not  usually  the  case  that  the  cost  per 
train-mile  for  any  small  additional  distance  is  as  high  as  the 
rate  already  given;    for   many  of  the  items,  such  as  station 
and  terminal  expenses,  which  go  to  make  up  the  average  cost 
per  train-mile,  are   not   affected   by  an  addition  in  distance 
which  does  not  exceed  2  or  3  p.  c.  of  the  total  length  of  the 
road.   Thus,  in  selecting  the  choice  of  two  routes,  the  engineer 


RAILROAD   LOCATION.  31 

should  not  necessarily  take  the  average  cost  per  train-mile  as 
his  standard  by  which  to  find  the  probable  difference  in  the 
operating-expenses,  but  in  most  cases  may  consider  about  50 
cents  per  train-mile  an  amply  sufficient  allowance  for  that 
portion  of  the  longer  route  which  is  in  excess  of  the  other, 
when  that  excess  does  not  exceed  the  above  amount. 

41.  In  order  to  approximate  as  closely  as  possible  to  the 
probable  cost  per  train-mile  on  any  projected  road,  the  en- 
gineer must  judge  by  the  results  on  other  roads  where  the 
conditions  are  more  or  less  similar.     Where  changes  are  to  be 
made  in  the  alignment  of  a  road  already  in  operation,  the 
value  of  the  proposed  improvements  can  then  be  found  with 
considerable  accuracy,  since  the  cost  per  train-mile  is  then 
known. 

BECEIPTS. 

42.  The  Receipts  usually  vary  from  about  1.5  to  2.0  the  cost 
of  operating;  and  it  is  not  often  that  the  locatiug-engineer  has 
it  in  his  power  to  affect  them  in  any  way.     He  may,  however, 
by  carrying  the  location  by  a  slightly  more  circuitous  route 
than  he  would  otherwise  have  adopted,  catch  the  traffic  of 
some  outlying  village.     Mr.  Wellington  on  this  subject  says: 
"  When  the  question  comes  up  of  lengthening  the  line  to 
secure  way-business,  we  may  almost  say  that  where  there 
seems  any  room  for  doubt,  it  will  almost  always  be  policy  to 
do  so.     Extra  business  to  a  railroad — the  engineer  will  rarely 
err  in  thinking — is  almost  always  clear  profit.     Of  Passenger 
business  this  is  literally  true  until  the  increase  becomes  con- 
siderable; of  Freight  business  it  is  so  nearly  true  that  80  or  90 
per  cent  at  least  of  the  way-rate  is  clear  profit  over  the  usual 
cost  of  any  particular  shipment." 

Thus,  suppose  we  are  projecting  a  line  between  two  points 
100  miles  apart,  and  that  half-way  between  them  lies  a  small 
town  10  miles  off  the  direct  route.  The  additional  distance 
involved  in  running  through  it  is  about  2  miles.  Suppose,  as 
is  a  reasonable  estimate,  the  average  payment  per  head  of  pop- 
ulation is  $13  per  annum.  Then,  if  there  are  likely  to  be  5 
daily  trains,  we  may  put  the  extra  cost  of  the  two  miles,  in- 
cluding the  interest  on  the  capital  spent  on  their  construction, 
at  about  f  2000  per  unuuua.  Therefore,  looking  at  the  matter 


32  RAILROAD   LOCATION". 

only  from  this  point  of  view,  if  the  place  contains,  or  is  likely 
to  contain  before  long,  only  about  150  people,  it  would  prob- 
ably be  wise  to  locale  the  road  through  it. 


COST  OF  CONSTRUCTION. 

43.  This  is  a  subject  which  had  almost  better  be  omitted, 
for  the  range  of  prices  is  so  great  in  different  parts  of  the 
country,  that  values  given  to  suit  one  place  may  be  entirely  mis- 
leading when  applied  to  another  place  a  few  hundred  miles  off. 
I  have,  however,  endeavored  to  strike  the  average  prices  as 
nearly  as  possible,  and  with  these  remarks  they  must  be  taken 
for  what  they  are  worth.  They  show  more  or  less  the  relative 
cost  of  various  works,  and  in  this  way  may  sometimes  be  of 
service. 

First  we  have  the  following  lot  common  to  all  track : 

Steel  rails  per  ton  (2000  Ibs.)... $25  00  to  $45  00 

Angle-bars,  per  Ib , 02  "         03 

Bolts,  "    03"          05 

Spikes,  " 02"         04 

Ties   (in  place),  each 20"         50 

Ballast— Gravel,  p.  cu.  yd 25  "         75 

"         Broken  Stone,  p.  cu.  yd 75"      150 

Track-laying  per  mile 250  00  "  500  00 

Then  we  have  the  following,  according  to  circumstances : 

Solid  Rock,  per  cu.yd ; $0  75  to  $  2  00 

Loose  Rock  or  Hard  Pan.  per  cu.  yd 35"  75 

Earth,  per  cu.  yd 10  "  50 

1st  Class  Masonry,  per  cu.  yd 10  00  "  30  00 

2d       "           "              "           ... 700"  1000 

3d        «'           "               "           , 500"  700 

Dry  rubble    "              "          200"  500 

Riprap,  per  cu.  yd 100"  200 

Iron  erected  in  bridge-work,  per  Ib ..  04"  OS 

Timber  in  Trestles,  per  M , 25  00  "  45  00 

"       "Culverts,      "    1500"  2500 

"       "  Log  Culverts,  per  M 1000"  2000 

Piling  driven,  per  lin.  ft  


Grubbing,  per  Station 1200 

Clearing,  per  acre 20  00 

Overhaul,  p.  cu.  yd.  per  Sta 01 

Fencing  per  mile  of  track 300  00 

Telegraph  line— Single  wire 175  00 


75 
20  00 
30  00 
02 

800  00 
250  00 


KA1LKOAD   LOCATION.  33 

By  taking  the  mean  prices  of  the  first  set,  we  obtain  for  an 
average  mile  of  standard-gauge  track  (10  p.  c.  short  rails)  the 
following  cost: 

103  tons  Steel  rails  (65  Ibs.  p .  yd.)  $3,862  00 

710  Angle-bars,  20  Ibs.  each 355  00 

1420  Bolts,  7  kegs,  200  Ibs.  each 56  00 

5670  Ibs.  Spikes,  38  kegs,  150  Ibs.  each 171  00 

2640  Ties    924  00 

Ballast,  3667  cu.  yds.  Gravel 1,834  00 

Track-laying  375  00 

Total $7,577  00 

Besides  these  we  have,  of  course,  Right  of  Way,  Engineer- 
ing, Law,  and  a  variety  of  Incidental  expenses. 

As  regards  the  COST  OF  TRESTLEWORK,  we  find  that  for  Low 
Pile  Trestles— say  20  ft.  high — assuming  piling  to  cost  50  cents 
per  lin.  ft.  driven,  and  the  superstructure  $20  per  M.,  the 
cost  will  usually  be  about  $6  per  foot  run. 

For  a  Wooden  Trestle  50  feet  high  at  $25  per  M.,  the  cost, 
if  resting  on  piles  or  sills,  will  usually  be  about  $10  per  foot 
run  ;  but  if  100  feet  high,  $20  to  $25  per  foot  run. 

The  cost  of  Iron  Trestlework  varies  so  enormously  accord- 
ing to  the  design,  that  it  is  impossible  to  lay  down  any  figures 
which  might  be  generally  applicable.  Assuming,  however, 
that  the  total  weight  of  iron  in  the  trestle  equals  the  total 
weight  of  wood  in  an  equally  strong  wooden  trestle,  the  cost, 
at  5  cents  per  lb.,  would  be  about  double  that  of  a  wooden 
one.  These  figures  are  of  course  exclusive  of  Masonry  foun- 
dations, and  are  for  single-track. 

As  regards  the  COST  OF  TRUSSES,  a  Wooden  Howe  Truss — 
single-track,  of  100  ft.  span,  Lumber  at  $15  per  M. — costs, 
framed,  somewhere  about  $2000;  and  an  Iron  Truss  of  the 
same  span,  at  5  cents  per  lb.,  costs  about  $5000.  The  cost  in 
both  cases  varies  pretty  much  as  the  square  of  the  span. 
Erecting  usually  costs  from  $5  to  $10  per  liu.  foot. 

As  regards  the  COST  OF  TUNNELLING,  we  may  say  it  varies 
from  $2.50  to  $7. 50 per  cu.  yd.;  so  that  for  a  single-track  tun- 
nel we  may  consider  the  price  per  foot  run  to  vary  from,  about 
$30  to  $80,  including  masonry.  The  cost  of  sinking  a  shaft  or 
driving  a  heading  is  considerably  higher  in  proportion  than  this. 

For  more  on  the  subject  of  the  Cost  of  Grading,  see  Sec. 
124,  Part  II. 


34  RAILROAD   LOCATION. 


INSTRUMENTS. 

44.  The  principal  Instruments  ordinarily  used  on  Railroad 
Location  are:  The  Transit,  Compass,  Level,  and  Hand  Level; 
and  we  will  consider  them  in  the  order  here  given .  (For  Instru- 
ments used  on  exploratory-work,  see  Sees.  141  to  158.) 


THE  TRANSIT. 

Before  proceeding  with  the  adjustments  of  the  Transit,  it 
should  be  seen  that  the  object-glass  is  screwed  firmly  home, 
and  a  short  scratch  made  on  the  ring  of  the  glass  and  contin- 
ued on  to  the  slide,  so  that,  should  the  glass  be  taken  out  or 
work  loose,  it  may  be  screwed  up  to  exactly  the  same  position 
it  was  in  before.  If  this  is  not  done,  and  the  glass  happens  to 
be  badly  centred, — i.e.,  its  optical  axis  does  not  lie  in  the  cen- 
tre of  the  telescope-tube, — if  by  any  chance  the  glass  is  moved, 
the  Line  of  Collimation  will  also  be  thrown  out  of  adjustment. 

The  following  are  the  usual  adjustments  for  a  Transit : 

A.  To  make  the  vertical  axis  truly  vertical  by  means 
of  the  small  bubble- tubes.     Turn  the  vernier- plate  until  each 
of  the  tubes  is  parallel  to  a  pair  of  opposite  plate-screws. 
Bring  both  bubbles  to  the  centres  of  the  tubes.     Then  turn 
the  instrument  through  about  180°.     If  the  bubbles  are  still  in 
the  centre,  the  adjustment  of  the  small  tubes  is  correct ;  but 
if  not,  correct  for  half  the  error  in  each  case  by  means  of  the 
adjusting  screws  at  the  ends  of  the  tubes.     This  adjustment 
should  then  be  correct ;  if  not,  repeat  the  process  until  it  is. 

B.  To  set  the  cross-hairs  truly  vertical  and  horizon- 
tal.— After  levelling  up,  test  the  vertical  hair  along  its  whole 
length  on  some  fixed  point,  and  if  not  correct,  loosen  the  cap- 
stan-headed screws  and  move  the  diaphragm  around.     The 
horizontal  hair  may  be  tested  in  a  similar  way. 

C.  To  make  the  horizontal  axis  of  the  telescope  truly 
horizontal. — Level  up  the  instrument  and  point  the  tele- 
scope to  some  object  C,  as  in  Fig.  5,  at  an  altitude,  if  possi- 
ble, of  not  less  than  45°.     Mark  the  point  A  where  this  verti- 
cal plane  strikes  the  ground.     "  Reverse"  the  instrument,  and 


KAILROAD   LOCATION.  35 

if  on  pointing  to  C  and  then  reducing  to  the  ground  we  again 
strike  A,   this  adjustment   is   correct.  c 

But  suppose  the  first  time  the  "verti- 
cal" plane  had  struck  the  ground  at  / 
B,  and  then  on  reversing,  instead  of                   / 
striking  B  again,  it  cuts  through  some                  ;' 
point  D.     Mark  a  point  E  between  D                I 
and  B,  distant  from  D  by  one  quarter  of                / 
DB.      Then  by  means  of  the  screws               / 
under  one  of  the  pivots  of  the  horizon-              / 
tal  axis  bring  the  intersection  of  the             /  £ 
cross-hairs  to  strike  the  point  E.     This  #»WW<M)««,Z«<< 
adjustment  should  then  be  correct.                       FlG- 5- 

D.  To  make  the  line  of  collimation  perpendicular 
to  the  horizontal  axis. — Having  levelled  up  the  instrument 
at  0,  in  Fig.  6,  point  the  telescope  to  some  object  (7.  Turn 
the  telescope  over  and  mark  the  point  A,  at  a  distance  AO 


-OC 


.E 
D"" 


FIG.  ,6. 


equal  to  about  OC,  where  it  strikes  the  ground  in  the  opposite 
direction.  By  making  AO  =  OC  we  then  obtain  a  correct 
adjustment  for  the  line  of  collimation,  even  though  the  object- 
slide  is  defective  ;  that  is  the  only  reason  for  making  AO  and 
OC  about  the  same  length.  Reverse,  and  again  point  to  C  \  if 
on  turning  the  telescope  over  once  more  it  again  strikes  A,  this 
adjustment  is  correct.  But  if  instead  of  intersecting  A  it 
cuts  through  some  other  point  D,  then  mark  a  point  E 
between  D  and  B,  distant  from  D  by  one  quarter  of  DB,  and 
by  means  of  the  capstan-headed  screws  move  the  diaphragm  so 
as  to  bring  the  intersection  of  the  cross-hairs  to  coincide  with 
E.  This  adjustment  should  then  be  correct.  This  is  liable  to 
throw  out  adjustment  B  slightly,  so  watch  that  at  the  same 
time. 

E.  To  make  the  long  bubble-tube  parallel  to  the  line 
of  collimation. — Level  up  the  instrument  and  clamp  the 
vertical  arc.  By  means  of  the  tangent-screw  of  the  vertical 
arc  bring  the  bubble  to  the  centre  of  the  tube.  Then  if  the 


36  BAILROAD   LOCATION. 

small  bubble-tubes  were  sufficiently  sensitive  to  render  the 
vertical  axis,  when  the  instrument  is  levelled  up,  truly  verti- 
cal, all  points  cut  by  the  line  of  collimation  equally  distant 
from  the  instrument  would  have  the  same  elevation.  But  it 
is  more  satisfactory  to  obtain  a  truly  vertical  axis  by  means  of 
the  long  bubble-tube  itself,  on  account  of  its  greater  sensitive- 
ness ;  thus :  Level  up  as  accurately  as  possible  by  the  small 
tubes,  and  then  treat  the  long  bubble-tube  as  if  it  were  one  of 
the  smaller  tubes,  putting  it  into  a  temporary  state  of  adjust- 
ment A,  by  means,  not  of  the  screws  at  the  ends  of  the  bubble- 
tube,  but  by  aid  of  the  tangent-screw  of  the  vertical  arc,  and 
then  by  its  means  obtain  a  truly  vertical  axis.  Then  take  the 
readings  on  two  points  A  and  B  equally  distant  from  the 
instrument  and  in  opposite  directions  ;  next  move  the  transit 
to  a  point  about  in  the  same  straight  line  as  A  and  B,  but  at  as 
short  a  distance  beyond  either  of  themjas  the  instrument  can 
be  focussed  to  read  and  level  up  by  the  small  tubes.  Take  the 
reading  at  A,  say  3.43  ;  then  if  B  were  previously  found  to  be 
1.84  feet  higher  than  A,  the  telescope  should  read  icSO'on  B 
if  this  adjustment  were  correct.  If  we  do  not  read  this,  the 
screws  at  the  end  of  the  long  bubble-tube  must  be  so  altered 
as  to  bring  the  bubble  to  the  centre  when  the  instrument  reads 
1.59.  On  again  pointing  to  A,  the  difference  between  A  and 
B  should  then  be  almost  1.59.  If  it  is  not  quite  1.59,  proceed 
as  before  until  the  adjustment  is  correct. 

By  moving  the  instrument  into  the  same  line  as  A  and  B,  as 
above,  we  avoid  the  necessity  of  levelling  up  this  vertical  axis 
again  by  means  of  the  long  bubble-tube. 

Besides  the  above  adjustments,  some  instruments  have  a  means 
of  Centring  the  Eye-piece  and  also  of  Adjusting  the  Object-Slide. 
(See  Note  C,  Appendix.) 

45.  Remarks.— Another  way  of  performing  adjustment  C 
is  by  means  of  an  object  and  its  reflection  in  still  water,  or 
even  in  a  plate  of  syrup.  A  star  at  night  does  well  for  this, 
but  it  is  advisable  to  select  one  as  nearly  east  or  west  as  pos- 
sible, as  its  motion  in  azimuth  is  then  a  minimum. 

If  at  any  time  adjustment  C  is  not  correct,  we  can  obtain 
true  results  by  "reversing,"  as  in  Fig.  5,  and  remembering  that 
half-way  between  the  two  points  so  found  is  the  correct  point. 

This  latter  remark  applies  also  to  adjustment  D.  It  is  a 
good  plan  to  reverse  on  a  back-sight  every  few  sights,  as  it 


RAILROAD   LOCATION".  37 

takes  practically  no  extra  time  and  at  once  detects  if  anything 
is  wrong.  By  taking  a  point  half-way  between  two  points,  as 
D  and  B  in  Fig.  6,  we  can  do  good  work  with  an  instrument 
in  which  this  adjustment  is  very  far  from  correct. 

As  regards  adjustment  E ': — If  we  had  a  level  handy,  it  is 
much  more  convenient  to  level  two  points  with  it ;  or  if  there 
is  a  sheet  of  still  water  at  hand,  two  pegs  driven  down  to  its 
surface  do  equally  well.  To  ascertain  the  Index-error  of  the 
vertical  circle  in  instruments  where  it  cannot  be  corrected  for 
instrumentally,  set  the  vertical  axis  truly  vertical,  as  explained 
under  adjustment  E,  then  level  up  the  telescope  and  observe 
the  readings  on  the  vertical  arc.  If  they  are  at  zero,  there  is  no 
index-error;  but  if  not,  the  difference  between  the  readings  and 
zero  is  the  index-error. 

If  the  transit  has  a  Striding -level  attached,  adjustment  C  may 
then  be  more  accurately  performed  by  means  of  it— whether 
the  striding-level  is  in  adjustment  itself  or  not,  for  it  is  only 
the  difference  of  the  readings  that  is  required.  To  make 
adjustment  C  then  proceed  thus:  Level  up  by  the  small  bubble- 
tubes  and  point  the  telescope  towards  the  north;  take  the  read- 
ings of  the  bubble  on  the  glass,  both  at  its  east  and  west  end; 
then  reverse  the  striding-level,  end  for  end,  and  take  the  read- 
ings a  second  time:  one  quarter  of  the  difference  between 
the  sum  of  the  two  east  readings  and  the  sum  of  the  two  west 
readings  equals  the  number  of  divisions  on  the  tube  that  the 
bubble  must  be  moved  by  means  of  the  pivot-screws  in  order 
to  make  the  "  horizontal  axis"  level,  that  end  being  too  high 
the  sum  of  whose  readings  is  the  greater.  If  the  striding-level 
is  in  adjustment,  we  have  only  to  screw  up  the  "horizontal 
axis"  so  as  to  agree  with  it.  We  can,  of  course,  adjust  the 
stridinff-level  by  placing  it  on  the  pivots  already  levelled,  and 
bringing  the  bubble  to  the  centre  of  the  tube. 

Lighting  the  cross-hairs,  when  the  instrument  has  no  lantern 
attached,  can  be  effected  by  fastening  a  piece  of  bright  tin — or 
even  white  paper— over  and  partly  in  front  of  the  object-glass, 
so  as  to  cast  the  reflection  of  a  light  on  the  ground  into  the  tube 
of  the  telescope;  but  the  reflector  must  not  obstruct  more  than 
half  of  the  field  of  the  object-glass.  A  piece  of  tin  or  paper 
with  a  i-inch  hole  in  the  centre  of  it,  fastened  at  a  suitable 
angle  over  the  object-glass,  answers  very  well. 
In  moving  the  diaphragm  when  the  telescope  has  an  invert- 


38  RAILROAD   LOCATION. 

ing  eye-piece,  it  has  to  go  in  the  opposite  direction  to  what  ap- 
pears to  be  the  right  one. 

If  working  with  an  instrument  the  graduation  of  which  is 
faulty,  read  each  angle  in  different  parts  of  the  circle.  The 
graduations  can  always  be  tested  by  reading  with  both  verniers 
on  various  parts  of  the  circle.  In  observing  an  angle,  if  we 
take  the  mean  result  obtained  by  both  verniers,  we  eliminate 
errors  due  to  eccentricity  of  the  vertical  axis  and  the  graduated 
circle,  as  well  as  reduce  the  errors  of  graduation. 

When  great  accuracy  is  required  in  reading  an  angle  the  best 
method  to  use  is  BORDA'S  REPETITION,  which  slightly  reduces 
the  errors  of  observation,  while  it  diminishes  those  of  gradua- 
tion in  in  verse  order  to  the  number  of  times  the  angle  is  re- 
peated. The  process  is  thus  :  Clamp  the  vernier-plate  to  zero, 
and  read  the  angle  by  both  verniers  according  to  the  usual 
method.  Then,  keeping  the  vernier-plate  clamped,  point  the 
telescope  again  to  the  first  object,  and  proceed  as  before  through 
any  number  of  repetitions.  At  the  end  of  the  final  angle  read 
the  verniers,  adding  360°  for  each  complete  revolution  which 
has  been  made,  and  divide  the  total  angular  measurement  by 
the  number  of  times  the  angle  was  repeated.  The  quotient  is 
the  required  angle.  In  this  way,  provided  there  is  no  play 
about  the  tangent-screws,  an  angle  can  be  read  with  confidence 
to  a  few  seconds  by  a  very  inferior  instrument. 

In  ordinary  work,  if  sure  of  the  correct  centring  of  the  verti- 
cal axis  and  also  of  the  graduation  itself,  there  is  no  need  to 
read  by  both  verniers;  but  it  is  advisable  to  read  always  by  the 
same  vernier  if  only  one  is  used. 

An  instrument  correct  according  to  the  adjustments  given 
above  gives  correct  results  when  dealing  with  objects  distant 
from  it  by  the  amount  OC  in  Fig.  6,  but  if  there  is  defective  cen- 
tring of  the  object- slide— not  to  be  confounded  with  eccentricity 
of  the  optical  axis  of  the  object-glass—it  will  not  give  correct 
results  in  dealing  with  objects  at  distances  from  it  greater  or  less 
than  OC.  This  can  always  be  tested  by  ranging  points  in  a 
" straight"  line  for  a  thousand  feet  or  so,  beginning  as  near  to 
the  instrument  as  the  focus  will  permit.  Then,  if,  on  ranging 
the  same  points  in  again  from  the  other  end,  they  do  not  coin- 
cide, one  half  the  difference  between  the  points  is  the  error 
in  alignment.  In  this  way,  even  with  a  bad  instrument,  a 
straight  line  can  be  run.  We  can  of  course  also  run  a  straight 


RAILROAD   LOCATIOK.  39 

line  with  an  instrument  which  has  a  defective  object-slide,  if 
in  proper  adjustment,  by  taking  back-sights  and  fore-sights 
equal  in  length  to  OC,  Fig.  6,  so  that  then  the  object-slide  will 
occupy  the  same  position  as  it  did  when  the  line  of  collimation 
was  adjusted. 

If  the  object-slide  works  correctly,  then,  although  the  object- 
glass  may  be  badly  centred, — i.e.,  its  optical  axis  will  not  co- 
incide with  the  centre  of  the  telescope- tube, — if  the  line  of  col- 
limation is  in  correct  adjustment  for  one  distance  it  will  give 
correct  results  at  all  distances. 

Parallax  is  caused  by  the  focus  of  the  object-glass  and  that 
of  the  eye-piece  not  coinciding  at  the  cross-hairs.  To  correct 
for  it,  shift  the  eye-piece  in  and  out  until  the  cross-hairs  are 
seen  distinctly.  Then  point  the  telescope  to  some  distant  object 
and  move  the  object-glass  in  and  out  until  the  image  of  the 
object  is  seen  sharp  and  clear,  coinciding  apparently  with 
the  cross-hairs. 

STADIA. 

46.  Transits  used  on  location  should  befitted  with  adjustable 
stadia-hairs.  These  are  usually  adjusted  to  read  1  foot  on  a  rod 
at  a  distance  from  the  centre  .of  the  instrument  equal  to  (100 
feet  -f-  distance  from  object-glass  in  its  mean  position  to  the 
centre  of  the  instrument  -f-  focal  length  of  the  object-glass), 
usually  making  a  distance  in  all  of  about  101.25  feet.  And 
since  the  stadia-hairs  should  be  placed  so  ias  to  be  equidistant 
from  the  ordinary  horizontal  hair,  at  a  distance  of  101.25  feet 
the  distance  read  between  each  pair  of  adjacent  hairs  should  be 
0.50  feet. 

If  the  hairs  are  not  adjustable,  but  are  fastened  to  the  oroSnary 
diaphragm,  then  the  measurements  on  the  rod  must  be  regulated 
to  suit  the  hairs,  remembering  that  the  .apex  of  the  angle  sub- 
tended by  the  distance  read  on  the  rod  is  not  at  the  centre  of  the 
instrument,  but  at  a  point  in  front  of  the  object-glass  by  a  dis- 
tance equal  to  the  focal  distance  of  the  object  glass,  which  is 
usually  1.25  feet  in  front  of  the  centre  of  the  instrument. 

If  the  hairs  are  unadjustable,  and  we  wish  to  use  an  ordi- 
nary levelling-rod  to  read  on,  then  suppose  at  101.25  feet  we 
read  0.88  feet  between  the  stadia-hairs,  we  must  divide  every 
reading  in  feet  by  0.88  in  order  to  obtain  the  distance  in  terms  of 


40 


KAILKOAD   LOCATION. 


100  feet.  Thus  if  at  a  certain  point  we  read  4.40  on  the  rod, 
the  distance  will  be  500  feet,  or  501.25  feet  from  the  centre  of 
the  instrument.  To  find  the  focal-length  of  the  object-glass, 
focus  it  for  a  distant  object;  the  distance  from  the  cross-hairs 
to  the  object-glass  then  equals  the  focal-length. 

On  sloping  ground,  if  the  rodman  is  careful  about  holding 
the  rod  perpendicular  to  the  line  of  sight,  swaying  it  slowly 
to  and  fro  so  as  to  permit  of  the  minimum  reading  being 
taken,  then  if  the  centre  hair  reads  somewhere  about  5  feet  on 
the  rod  (i.e.,  the  height  of  the  instrument  above  the  ground) 
we  have  only  to  multiply  the  distance  as  read  on  the  incline 
by  the  cosine  of  the  inclination,  in  order  to  obtain  the  true 
horizontal  distance. 

But  if  really  correct  work  is  wanted,  it  is  best  to  have  a  bub- 
ble-tube attached  to  the  rod  so  that  it  can  be  held  vertically, 
and  then  correct  for  the  inclination  as  follows  : 


FIG.  7. 
In  Fig.  7  the  distance 

EF=ABcos*  FEC, 

EF being  in  terms  of  100  feet,  and  PEG  being  the  angle  of 
inclination  as  measured  to  C,  the  ordinary  horizontal  hair  of 
the  instrument,  assuming  that  the  stadia-hairs  are  equidistant 
from  C,  and  that  1  foot  on  the  rod  corresponds  with  100  feet 
in  distance. 

In  order  to  reduce  this  to  the  centre  of  the  instrument,  we 
should  of  course  add  to  EFthe  amount  1.25  X  cos  FEC,  but 
for  ordinary  inclinations  we  may  assume  this  correction  to 
equal  1  foot.  Thus,  if  FEC  =  30°,  and  AB  =  6.00,  then 


RAILROAD   LOCATION". 


41 


E'F-  1  ft.  -f  (6  X  .75)=  451  feet.  To  obtain  the  height  HG 
in  Fig.  7,  the  best  way  is  to  make  CH  on  the  rod  equal  to  the 
height  of  the  point  E  above  the  ground,  say  5  feet.  Then 

HG  =  EF  tan  FEG. 

Thus  in  the  above  example  HG  =  260  feet.  The  following 
table  gives  the  VALUES  OF  COS2  FEG,  where  FEG  is  the 
inclination  angle: 


INCLINA- 
TION. 

0' 

10' 

20' 

30' 

40' 

50' 

0° 

1.0000 

1.0000 

1.0000 

.9999 

.9999 

.9998 

1 

.9997 

.9996 

.9995 

.9993 

.9992 

.9990 

2 

.9988 

.9986 

.9983 

.9981 

.9978 

.9976 

3 

.9973 

.9969 

.9966 

.9963 

.9959 

.9955 

4 

.9951 

.9947 

.9943 

.9938 

.9934 

.9929 

5 

.9924 

.9919 

.9914 

.9908 

.9902 

.9897 

6 

.9891 

.9885 

.9878 

.9872 

.9865 

.9858 

.9851 

.9844 

.9837 

.9830 

.9822 

.9814 

8 

.9806 

.9798 

.9790 

.9782 

.9773 

.9764 

9 

.9755 

.9746 

.9737 

.9728 

.9718 

.9708 

10 

.  .9698 

.9688 

.9678 

.9668 

.9657 

.9647 

11° 

.9636 

.9625 

.9614 

.9603 

.9591 

.9580 

12 

.9568 

.9556 

.9544 

.9532 

.9519 

.9507 

13 

.9494 

.9481 

.9^68 

.9455 

.9442 

.9428 

14 

.9415 

.9401 

.9387 

.9373 

.9359 

.9345 

15 

.9330 

.9315 

.9301 

.9286 

.9271 

.9256 

16 

.9240 

.9225 

.9209 

.9193 

.9177 

.9161 

17 

.9145 

.9129 

.9112 

.9096 

.9079 

.9062 

18 

.9045 

.9028 

.9011 

.8993 

.8976 

.8958 

19 

8940 

.8922 

.8904 

.8886 

.8867 

.8849 

20 

.8830 

.8811 

.8793 

.8774 

.8754 

.8735 

21° 

.8716 

.8696 

.8677 

.8657 

-  .8637 

.8617 

22 

.8597 

.8576 

.8556 

.8536 

.8515 

.8494 

23 

.8473 

.8452 

.8431 

.8410 

.8389 

.8367 

24 

.8346 

.8324 

.8302 

.8280 

.8258 

.8236 

25 

.8214 

.8192 

.8169 

.8147 

.8124 

.8101 

26 

.8078 

.8055 

.8032 

.8009 

.7986 

.7962 

27 

.7939 

.7915 

.7892 

.7868 

.7844 

.7820 

28 

.7796 

.  777'' 

.7747 

.7723 

.7699 

.7674 

29 

.7650 

.7625 

.7600 

.7575 

.7550 

.7525 

30 

.7500 

.7475 

.7449 

.7424 

.7398 

.7373 

31° 

.7347 

.7322 

.7296 

.7270 

.7244 

.7218 

32 

.7192 

.7166 

7139 

.7113 

.7087 

.7060 

33 

.7034 

.7007 

.6980 

.6954 

.6927 

.6900 

34 

.6873 

.6846 

.6819 

.6792 

.6765 

.6737 

35 

.6710 

.6683 

.6655 

.6628 

.6600 

.6573 

36 

.6545 

.6517 

.6490 

.6462 

.6434 

.6406 

37 

.6378 

.6350 

.6322 

.6294 

.6266 

.6238 

38 

.6210 

.6181 

.6153 

.6125 

.6096 

.6068 

39 

.6039 

.6011 

.5982 

.5954 

.5925 

.5897 

40 

.5868 

.5839 

.5811 

.5782 

.5753 

.5725 

m 

42  RAILROAD   LOCATION. 


THE  COMPASS. 

47.  The  adjustments  of  the  Compass  are  as  follows: 

A.  To  make   the  needle  swing  horizontally.— Level 

the  compass,  then  by  means  of  the  slide-piece  on  the  needle 
regulate  its  centre  of  gravity  so  that  it  will  swing  horizontally. 

B.  To  straighten  the  needle.— See  if  both  ends  of  the 
needle  point  to  exactly  opposite  graduations  while  the  compass 
is  being  turned  completely  around.  If  so,  the  needle  is  straight 
and  the  pivot  is  properly  centred.     But  if  not,  the  error  will 
arise  from  either  one  or  both  of  these  not  being  correct.    Turn 
the  compass  until  some  graduation,  say  90°,  comes  precisely 
to  the  northern  end  of  the  needle.     Mark  the  place  where  the 
southern  end  of  the  needle  then  points.     Take  off  the  needle 
and  bend  it  until  its  southern  end  points  half-way  between  90° 
and  the  point  already  marked,  while  its  northern  end  is  kept 
at  the  opposite  90°  by  slightly  moving  the  compass  around. 
The  needle  will  then  be  straight,  although  it  will  not  intersect 
opposite  degrees  on  account  of  the  eccentricity  of  the  pivot. 

C.  To  centre  the  pivot. — Turn  the  compass  around  until 
a  place  is  found  where  the  opposite  ends  of  the  needle  cut 
opposite  degrees.  Then  turn  the  compass  quarter- way  around, 
or  through  90°.     If  the  needle  then  cuts  opposite  degrees,  the 
pivot  is  in  adjustment;  but  if  not,  bend  the  pivot  until  it  does. 
The  needle  should  then  cut  opposite  degrees  while  being 
turned  completely  around. 

Remarks. — If  the  magnetism  of  the  needle  gets  weak,  it 
may  be  renewed  as  follows:  Cover  the  needle  with  a  thin  film 
of  oil,, and  then  with  the  north  pole — the  end  marked  with  a 
line  across  it— of  an  ordinary  magnet  rub  the  south  end  of 
the  needle,  beginning  at  the  centre  and  working  outwards 
towards  the  end;  similarly  rub  the  north  end  of  the  needle 
with  the  south  pole  of  the  magnet.  After  doing  this  a  few 
times  the  magnetism  should  be  sufficiently  restored. 

Reading  both  ends  of  the  needle  corrects  for  eccentricity  of 
the  pivot  if  the  needle  is  straight;  it  also  of  course  reduces  the 
errors  of  graduation. 

Should  the  glass  cover  become  electrified,  as  it  will  if  but 
slightly  rubbed,  so  that  the  needle  sticks  to  the  under  side  of 
it  and  will  not  "traverse"  properly,  touching  the  glass  in 


RAILROAD   LOCATION.  43 

several  places  with  the  moistened  finger,  or  breathing  on  it, 
will  remove  the  electricity. 

A  compass  when  left  standing  for  any  considerable  time 
should  always  have  its  needle  free,  in  order  to  prevent  loss  of 
magnetic  power.  Of  course  when  carried  it  should  always  be 
clamped. 

In  taking  a  compass-reading,  not  only  must  all  iron  and 
steel  substances  be  kept  well  away,  but  metal  magnifiers  of  all 
sorts  are  liable  to  cause  a  slight  deflection,  owing  to  the  possi- 
bility of  impurity  in  the  material  of  which  they  are  composed. 
Magnifiers  coated  with  nickel  are  especially  bad,  since  nickel 
itself  is  a  decidedly  magnetic  metal. 

Since  the  magnetic  attraction  varies  in  different  places, 
adjustment  A,  if  correct  in  one  place,  will  probably  want 
looking  to  if  the  instrument  is  taken  anywhere  else. 

MAGNETIC  VARIATION. 

48.  By  referring  to  the  Chart  of  Magnetic  Variation,  we  see 
that  in  North  America  the  variation  is  both  towards  the  east 
and  the  west.  The  "  line  of  no  variation"  which  separates 
these  two  divisions  is  found  to  be  constantly  shifting  west- 
wards at  an  average  rate  of , about  4'  per  annum.  This  causes 
a  gradual  increase  in  all  variations  to  the  west,  and  a  corre- 
sponding decrease  in  all  variations  to  the  east ;  and  changes 
similar  to  these  are  going  on  all  over  the  globe.  Besides  this 
secular  variation  we  have  diurnal  and  annual  variations,  but 
for  practical  field  purposes  these  latter  may  be  ignored.  The 
former  of  them  is  such  that  the  needle  attains  its  extreme 
westerly  position  at  about  2  P.M.  each  day,  and  its  extreme 
easterly  position  at  about  8  A.M.;  while  the  latter  shows  itself 
generally  by  a  slight  increase  in  variations  west,  and  decrease 
in  variations  east,  during  the  summer.  - 

The  chart  here  given  is  more  as  a  matter  of  interest  than 
for  any  real  use  in  the  field.  If  the  variation  at  any  place  is 
wanted  accurately,  usually  the  only  satisfactory  way  is  to  take 
it  directly  by  observation,  as  shown  in  Sec.  57.  For  very 
rough  work,  however,  an  idea  of  the  amount  of  variation  can 
be  obtained  from  the  chart  by  interpolating  by  eye. 

The  "lines  of  no  variation"  are  shown  thicker  than  the 
others. 


8        8 


8       8 


8        3 


BAILROAD   LOCATION".  45 


THE  LEVEL. 

49.  We  will  first  take  the  DUMPY  LEVEL,  which  usually 
needs  only  two  adjustments. 

A.  To  make  the  bubble-tube   perpendicular  to  the 
vertical  axis. — This  is  done  in  just  the  same  way  as  with 
one  of  the  small  bubble-tubes  in  adjustment  A  of  the  transit. 

B.  To  make  the  line  of  collimation  parallel  to  the 
bubble-tube. — This  is  done  in  a  similar  way  to  the  adjust- 
ment of  the  long  bubble-tube  of  a  transit,  already  described, 
except  that  in  this  case  it  is  the  line  of  collimation  that  has  to 
be  made  parallel  to  the  bubble-tube,  so  that  now  it  is  the 
cross-hairs  that  have  to  be  moved.     In  this  case  of  course  there 
is  no  necessity  to  set  up  the  instrument  "  in  about  the  same 
line  as  A  and  B"  as  there  was  in  the  case  of  the  transit. 

Another  way  of  performing  this  adjustment  is  by  the 
method  of  "  reciprocal  observations,"  as  given  for  the  Hand- 
level  in  Sec.  52. 

The  remarks  which  applied  to  the  telescope  of  a  transit  ap- 
ply with  equal  force  to  the  telescope  of  a  level ;  more  especially 
the  remark  on  the  running  of  a  straight  line  if  the  object-slide 
is  badly  centred.  If  the  level  has  a  means  of  adjusting  the 
eye-piece  and  object-slide,  see  Note  C,  Appendix. 

50.  The  Y  Level  has  three  adjustments  as  follows: 

A.  To   make  the  line  of  collimation   coincide  with 
the  axis  of  the  telescope. — Open  the  clips  of  the  Y's.    To 
adjust  the  vertical  hair,  mark  the  intersection  of  the  cross-hairs 
on  some  fixed  object,  and  revolve  the  telescope  in  its  Y's  so 
that  the  level  will  be  upside  down;  and  then  if  the  intersection 
falls  to  one  side  of  the  object,  one  half  the  error  must  be  cor- 
rected for  by  the  capstan -headed  screws.     To  adjust  the  hori- 
zontal hair,  turn  the  telescope  over  as  before,  and  if  the  inter- 
section of  the  hairs  strikes  above  or  below  the  object,  correct  as 
before  for  one  half  the  error. 

B.  To  make  the  bubble-tube  parallel  to  the  line  of 
collimation. — This  adjustment  consists  of  two  parts.    First, 
bring  the  bubble  to  the  centre  and  then  revolve  the  telescope 
in  its  Y's  through  about  20° ;  if  the  bubble  then  runs  to  one  end, 
half  the  error  must  be  corrected  for  by  the  horizontal  screws 
at  the  end  of  the  tube,  raising  or  lowering  as  may  be  required. 


46  RAILROAD   LOCATION-. 

For  the  second  part  of  this  adjustment,  place  the  telescope  over 
a  pair  of  opposite  levelling-screws,  open  the  clips,  and  bring 
the  bubble  to  the  centre  of  the  tube.  Reverse  the  telescope 
end  for  end  in  its  Y's;  if  the  bubble  is  not  then  in  the  centre, 
one  half  the  error  must  be  corrected  for  by  the  vertical  screws 
at  the  end  of  the  bubble-tube.  On  levelling-up  and  again  re- 
versing, this  adjustment  should  be  found  to  be  correct. 

C.  To  make  the  axis  of  the  telescope  perpendicular 
to  the  vertical  axis. — Level  up.  Place  the  telescope  over 
a  pair  of  opposite  levelling-screws.  Swing  the  telescope  half- 
way round  on  its  vertical  axis.  If  then  the  bubble  has  left 
the  centre,  bring  it  half-way  back  by  means  of  the  large  cap- 
stan-headed nuts  of  the  Y's.  Then  place  the  telescope  over 
the  other  pair  of  levelling-screws,  and  if  necessary  proceed  as 
before.  This  adjustment  should  then  be  correct. 

Remarks.— As  with  the  transit,  if  the  object-slide  of  a  level 
is  defective  the  line  of  collimation  when  adjusted  is  only  cor- 
rect for  back-sights  and  fore-sights  of  equal  length  with  the 
distance  of  the  object  on  which  the  line  of  collimation  was 
adjusted. 

In  levelling,  whenever  possible,  keep  the  fore-sights  and 
back-sights  of  equal  length:  if  so,  accurate  work  can  be  done 
with  an  instrument  thoroughly  out  of  adjustment,  for  then  the 
actual  height  of  the'  instrument  itself  is  of  no  importance.  If, 
as  in  levelling  uphill,  it  is  necessary  to  take  extremely  short 
fore-sights,  they  should  be  counteracted  by  short  back  sights 
— if  not  at  the  time,  as  soon  afterwards  as  possible. 

51.  There  is  no  need  to  allow  for  CURVATURE  OF  THE  EARTH 
OR  REFRACTION  in  sights  under  700  feet,  and  then,  if  taking 
fore-sights  and  back-sights  of  about  the  same  length,  the 
corrections  would  counteract  each  other;  so  that  it  is  only  in 
taking  an  extremely  long  fore-sight  or  back-sight,  which  is 
not  counteracted  by  a  more  or  less  equal  sight  in  the  opposite 
direction,  that  we  need  apply  corrections  for  curvature  or 
refraction. 

For  CURVATURE  the  correction  in  feet  amounts  to 

0.67J&9, 

where  L  =  length  of  sight  in  miles,  and  is  to  be  subtracted 
from  the  reading  on  the  rod:  this  being  simply  the  tangential 


RAILROAD   LOCATION. 


offset  for  a  curve — see  Sec.  78— with  radius  equal  that  of  the 
earth. 
For  REFRACTION,  on  an  average,  it  amounts  in  feet  to 

0.1L2, 

which  is  an  experimental  quantity,  and  is  to  be  added  to  the 
reading  on  the  rod.  So  that,  taking  the  twTo  together,  we  may 
say  that  the  correction  in  feet  amounts  to 

0.57Z'2, 

and  is  to  be  subtracted  from  the  reading  on  the  rod,  the  eleva- 
tion as  taken  or  given  by  the  level  being  always  too  low. 
This  is  equivalent  to  about  .0002  ft.  at  100  feet ;  so  that,  since 
it  increases  as  the  square  of  the  distance  at  say,  1200  feet,  it 
will  equal  .0002  X  122  =  .03  foot.  The  following  table  gives 

the  JOINT  CORRECTIONS  FOR  CURVATURE  AND  REFRACTION, 

worked  out  by  the  above  formula,  and  is  useful  in  ascertain- 
ing the  elevation  of  the  surrounding  country  : 


Distance  in  Miles. 

Correction  in  Ft. 

Distance  in  Miles. 

Correction  in  Ft. 

1 

0.57 

30 

513 

5 

14.25 

40 

912 

10 

57.0 

50 

1,425 

15 

128 

60 

2,052 

20 

228 

80 

3,648 

25 

356 

100 

5,700 

Thus  from  the  table,  if  the  level  gives  a  point  on  a  distant 
mountain,  say  30  miles  off,  the  elevation  of  that  point  will  be 
equal  to  the  elevation  of  the  instrument  -j-  513  feet. 

52.  The  Hand-Leyel. — The  only  adjustment  necessary  as 
a  rule  with  this  instrument  is  to  make  the  line  of  collimation 
parallel  to  the  bubble-tube.  To  do  this,  sight  from  a  point  A 
to  a  point  B,  as  in  Fig.  8,  and  then  back  again  from  B  to  A. 


B 


C 


FIG.  8. 


If  the  level  is  in  adjustment  the  two  sights  should  coincide  at 
A.    But  suppose  C  is  the  point  struck  instead  of  A  ;  then  D, 


48  EAILEOAD   LOCATION. 

a  point  half-way  between  A  and  0,  will  be  on  a  level  with  B : 
therefore  the  hair  must  be  adjusted  on  the  line  BD.  A 
handier  way  is  of  course  to  adjust  it  by  means  of  another  level, 
or  a  sheet  of  water. 


THE  SUBYEY. 

53.  The  object  of  the  following  notes  is  not  to  show  the 
mode  of  conducting  location, — which  of  course  can  only  be 
picked  up  by  actual  experience  in  the  field,— but  merely  to 
give  solutions  of  the  various  mathematical  and  instrumental 
problems  which  arise  in  the  course  of  the  work. 

In  the  case  of  Exploratory  Surveys,  the  instruments  used  and 
the  problems  which  arise  being  usually  entirely  different  from 
those  which  come  into  question  in  ordinary  location,  they  will 
be  considered  separately  in  Part  III. 

A  Reconnoissance  survey,  as  generally  understood,  may  also 
be  classed  with  the  above,  or  it  may  take  the  form  of  a  rough 
preliminary  survey,  a  compass  perhaps  being  substituted  for 
the  ordinary  transit.  As  regards  Compass-surveys,  there  is 
among  engineers  a  strong  prejudice  against  them,  but  in  a 
country  tolerably  free  from  local  attraction  a  compass-line  is 
surely  correct  enough  for  preliminary  work  ;  for  though  by  it 
accuracy  cannot  be  obtained  at  any  one  point,  its  errors  are 
not  accumulative,  but  in  a  great  measure  counteract  each 
other,  so  that  the  line  as  a  whole  should  give  very  fair  results. 
Another  method  of  performing  rough  wrork  is  the  Stadia 
process,  by  means  of  which  very  good  results  have  often  been 
obtained,  the  engineering  staff  consisting  merely  of  an  engi- 
neer and  a  rodman,  the  only  instruments  used  being  a  rod  and 
a  small  transit  with  bubble-tube  and  stadia  attachment.  Com- 
paring compass  and  stadia  work,  the  former  is  usually  more 
suitable  in  timber  and  the  latter  in  open  country. 

The  term  Preliminary  survey  is  variously  used,  sometimes 
indicating  a  mere  reconnoissance,  but  more  generally  a 
survey  the  object  of  which  is  to  obtain  accurate  topography, 
in  order  by  its  means  to  select  the  final  location.  As 
regards  the  degree  of  accuracy  to  be  employed  in  preliminary 
work,  it  of  course  depends  in  what  way  the  results  are  to  be 


RAILROAD  LOCATION".  49 

used  ;  but  it  is  generally  best  to  run  the  transit-line  of  the 
"  preliminary"  with  as  much  accuracy  as  is  attainable  under 
the  circumstances.  If  this  is  done  we  then  have  a-  line  on 
which  we  can  at  all  points  depend,  and  which  we  can  use  as 
a  base  for  other  lines,  knowing,  if  we  branch  off  from  it  at 
one  point,  the  exact  course  we  must  make  to  strike  it  again  at 
any  given  station.  We  will  therefore  suppose  in  the  follow- 
ing notes  that  the  final  location  is  to  be  selected  by  the  aid  of 
an  accurately  run  preliminary  line,  topography  having  been 
taken  on  either  side  of  the  transit  line  to  a  distance  of  from, 
say,  100  to  600  feet,  according  to  the  nature  of  the  ground. 

On  Prelimin.iry  Surveys,  by  means  of  a  hand-level  and 
prismatic-compass,  ihe  engineer-in-charge,  keeping  ahead  of 
the  party,  is  generally  able  to  ascertain  approximately  where 
the  line  will  go,  and  then  the  transit-man  has  merely  to  follow 
more  or  less  the  route  indicated,  being  guided  by  the  con- 
sideration of  running  the  line  as  much  as  possible  to  a  con- 
stant rate  of  grade.  If  the  line,  however,  is  being  run  to  the 
maximum  grade— or  any  other  rate  of  grade  which  it  is  the 
wish  of  the  engineer  to  maintain — along  a  continuous  trans- 
verse slope,  such  as  a  mountain-side,  the  transit-man  can 
choose  the  line  tolerably  well  for  himself,  since  he  only  has  to 
select  his  stations  so  as  to  maintain  the  required  rate,  which 
he  can  do  by  means  of  the  vertical  arc.  But  in  selecting  these 
points  he  has  to  bear  in  mind  the  probable  amount  of  curva- 
ture which  there  will  be  between  the  station  where  the  instru- 
ment is  standing  and  the  place  at  which  the  front  picket  is  to 
be  set,  and  allow  for  it  in  setting  the  picket.  (See  Sec.  26.) 
Thus,  suppose  he  is  running  the  line  to  a  1.5  p.  c.  grade,  and 
that  he  estimates  the  distance  to  the  picket  to  be  about  500 
feet,  and  the  probable  total  curvature  in  that  distance  to  be 
15°,  then  the  grade-angle,  instead  of  being  51|',  as  in  the  fol- 
lowing table,  will  be  48^'.  If  he  has  stadia-hairs  in  his  instru- 
ment,— as  he  ought  to  have,— he  can  read  off  the  distance  with 
sufficient  accuracy  on  the  picket  itself,  and  in  this  way  form 
his  estimates  more  closely.  The  difference  in  distance  along 
the  straight  course  and  along  the  probable  location  must  also 
be  allowed  for  where  the  deviation  is  great.  The  following 
is  a 


50 


KAILROAD   LOCATION. 


TABLE  OF  GRADES  AND  GRADE-ANGLES. 


Feet  perl 
Station. 

Feet  pei- 
Mile. 

Inclina- 
tion. 

Feet  pei- 
Station. 

Feet  per 
Mile. 

Inclina- 
tion. 

Feet  per 
Station. 

Feet  pei- 
Mile. 

Inclina- 
tion. 

0       1       II 

0        /        // 

0        /        // 

.01 

.528 

21 

.51 

26.928 

17  32 

1.01 

53.328 

31  43 

.02 

1.056 

41 

.52 

27.456 

17  53 

|l.02 

53.856 

35  04 

.03 

1.584 

1  02 

.53 

27.984 

18  13 

.03 

54.384 

35  24 

.04 

2.112 

1  23 

.54 

28.512 

18  34 

.04 

54.912 

35  45 

.05 

2.640 

1  43 

.55 

29.040 

18  54 

.05 

55.440 

36  05 

.06 

3.168 

2  04 

.56 

29.568 

19  15 

.06 

55.968 

36  26 

.07 

3.696 

224 

.57 

30.096 

19  86 

.07 

56.496 

36  47 

.08 

4.224 

2  45 

.58 

30.624 

19  5G 

.08 

57.024 

37  08 

.09 

4.752 

3  06 

.59 

31.152 

20  17 

.09 

57.552 

37  28 

.10 

5.280 

3  26 

.CO 

31.680 

20  38 

.10 

58.080 

37  49 

.11 

5.808 

347 

.61 

32.208 

20  58 

.11 

58.608 

38  09 

.12 

6.336 

408 

.62 

32.736 

21  19 

.12 

59.136 

38  30 

.13 

6.864 

4  28 

.63 

33  264 

21  39 

.13 

59.664 

38  51 

.14 

7.392 

4  49 

.64 

33.792 

22  00 

.14 

60.192 

39  11 

.15 

7.920 

5  09 

.65 

34.320 

22  21 

.15 

60.720 

39  32 

.16 

8.448 

5  30 

.66 

34.848 

22  41 

!   .16 

61.248 

39  53 

.17 

8.976 

5  51 

.67 

35.376 

23  02 

i   .17 

61.776 

40  13 

.18 

9.504 

6  11 

.68 

35.904 

23  23 

;   .18 

62.304 

40  34 

.19 

10.032 

6  32 

.69 

36.432 

23  43 

\   .19 

62.832 

40  54 

.20 

10.560 

6  53 

.70 

36.960 

24  04 

.20 

63.360 

41  15 

.21 

11.088 

7.13 

.71 

37.488 

24  24 

.21 

63.888 

41  35 

.2-2 

11.616 

7  34 

.72 

38.016 

24  45 

.22 

64.416 

41  56 

.23 

12.144 

7  54 

.73 

38.544 

25  06 

1   .23 

64.944 

42  17 

.24 

12.672 

8  15 

.74 

39.072 

25  26 

.24 

65.472 

42  38 

.25 

13.200 

8  36 

.75 

39.600 

25  47 

.25 

66.000 

42  58 

.26 

13.728 

8  56 

.76 

40.128 

26  08 

.26 

66.528 

43  19 

.27 

14.256 

9  17 

.77 

40.656 

26  28 

.27 

67.056 

43  39 

.28 

14.784 

9  38 

.78 

41.184 

26  49 

'    .28 

67.584 

44  00 

.29 

15.312 

9  58 

.79 

41.712 

27  09 

.29 

68.112 

44  21 

.30 

15.840 

10  19 

.80 

42.240 

27  30 

.30 

68.640 

44  41 

.31 

16.368 

10  39 

.81 

42.768 

27  51 

.31 

69.168 

45  02 

.32 

16.896 

11  00 

.82 

43.296 

28  11 

i   .32 

69.696 

45  23 

.33 

17.424 

11  21 

.83 

43.824 

28  32 

.33 

70.224 

45  43 

.34 

17.952 

11  41 

.84 

44.352 

28  53 

.34 

70.752 

46  04 

.35 

18.480 

12  02 

.85 

44.880 

29  13 

.35 

71.280 

46  24 

.36 

19.008 

12  23 

.86 

45.408 

29  34 

.36 

71.808 

46  45 

.37 

19.536 

12  43 

.87 

45.936 

29  54 

.37 

72.336 

47  06 

.38 

20.064 

13  04 

.88 

46  464 

30  15 

.38 

72.864 

47  26 

.39 

20.592 

13  24 

.89 

46.992 

30  36 

.39 

73.392 

47  47 

.40 

21.120 

13  45 

.90 

47.520 

30  57 

.40 

73.920 

48  08 

.41 

21.648 

1406 

.91 

48.048 

31  17 

.41 

74.448 

48  28 

.42 

22.176 

14  26 

.92 

48.576 

31  38 

i   .42 

74.976 

48  49 

.43 

22.704 

14  47 

.93 

49.104 

81  58 

.43 

75.504 

49  09 

.44 

23.232 

15  08 

.94 

49.682 

32  19 

.44 

76.032 

49  30 

.45 

23.760 

15  28 

.95 

50.160 

32  39 

.45 

76  560 

49  51 

.46 

24.288 

15  49 

.96 

50.688 

33  00 

I   .46 

77.088 

50  11 

.47 

24.816 

16  09 

.97 

51.216 

33  21 

.47 

77.616 

50  32 

.48 

25.344 

16  30 

.98 

51.744 

33  41 

!    .48 

78.144 

50  52 

.49 

25.872 

16  51 

.99 

52.272 

34  02 

.49 

78.672 

51  13 

.50 

26.400 

17  11 

1.00 

52.800 

34  23 

.50 

79.200 

51  34 

RAILROAD   LOCATION. 


51 


TABLE    OF  GRADES  AND  GRADE-ANGLES.— Continued. 


35c 
2k2 

11 

Feet  per 
Mile. 

Inclina- 
tion. 

OJ  S3 

2% 

4)  03 

£% 

Feet  per 
Mile. 

Inclina- 
tion. 

1  Feet  per 
|  Station 

Feet  per 
Mile. 

Inclina- 
tion. 

0       /       // 

0       /       // 

0       /       '/ 

.51 

79.728 

51  54 

1.91 

100.848 

1  05  39 

3.55 

187.440 

2  01  59 

.52 

80.256 

52  15 

1.92 

101.376 

1  06  00 

3.60 

190.080 

2  03  42 

.53 

80.784 

52  36 

1.93 

101.904 

1  06  20 

3.65 

192.720 

2  05  25 

.54 

81.312 

52  56 

1.94 

102.432 

1  06  41 

3.70 

195.360 

2  07  08 

.55 

81.840 

53  17 

1.95 

102.960 

1  07  02 

3.75 

198.000 

2  08  51 

.56 

82.368 

53  37 

1.96 

103.488 

1  07  22 

3.80 

200.640 

2  10  34 

.57 

82.896 

53  58 

1.97 

104.016 

07  43 

3.85 

203.280 

2  12  17 

.58 

83.424 

54  19 

1.98 

104.544 

08  04 

3.90 

205.920 

2  14  00 

.59 

83.952 

54  39 

1.99 

105.072 

08  24 

3.95 

208.560 

2  15  43 

.60 

84.480 

55  00 

2.00 

105.600 

08  45 

4.00 

211.200 

2  17  26 

.61 

85.008 

55  21 

2.05 

108.240 

10  28 

4.10 

216.480 

2  20  52 

.62 

&5.536 

55  41 

2.10 

110.880 

12  11 

4.20 

221.760 

2  24  18 

.63 

86  064 

56  02 

2.15 

113.520 

13  54 

4.30 

227.040 

2  27  44 

.64 

86.592 

56  22 

2.20 

116.160 

15  37 

4.40 

232.320 

2  31  10 

.65 

87.120 

56  43 

2.25 

118.800 

17  20 

4.50 

237.600 

2  34  36 

66 

87.648 

57  04 

2.30 

121.440 

19  03 

4.60 

242.880 

2  38  01 

.67 

88.176 

57  24 

2.35 

124.080 

20  46 

4.70 

248.160 

2  41  27 

.68 

88.1  04 

57  45 

2.40 

126.720 

2229 

4,80 

253.440 

2  44  53 

.69 

89.232 

58  06 

2.45 

129.360 

24  12 

4.90 

258.720 

2  48  19 

.70 

89.760 

58  26 

2.50 

132.000 

25  56 

5.00 

264.000 

2  51  45 

.71 

90.288 

58  47 

2.55 

134.640 

1  2739 

5.10 

269.280 

2  55  10 

.72 

90.816 

59  07 

2.60 

137.280 

1  29  22 

5.20 

274.560 

2  58  36 

.73 

91.344 

59  28 

2.65 

139.920 

1  31  05 

5.30 

27*9.840 

3  02  09 

.74 

91.872 

59  49 

2.70 

142.560 

1  32  48 

5.40 

285.120 

3  05  27 

.75 

92.400 

1  00  09 

2.75 

145.200 

1  34  31 

5.50 

290.400 

3  08  53 

.76 

92.928 

1  00  30 

2.80 

147.840 

1  36  14 

5.60 

295.680 

3  12  19 

.77 

93.456 

1  00  51 

2.85 

150.480 

1  37  57 

5.70 

300.960 

3  15  44 

78 

93.984 

1  01  11 

2.90 

153.120 

1  39  40 

5.80 

306.240 

3  19  10 

.79 

94.512 

1  01  32 

2.95 

155.760 

1  41  23 

5.90 

311.520 

3  22  36 

80 

95.040 

1  01  52 

3.00 

158.400 

1  43  06 

6.00 

316.800 

3  26  01 

.81 

95.568 

1  02  13 

3.05 

161.040 

44  49 

6.10 

322.080 

3  29  27 

.82 

96.096 

1  02  34 

3.10 

163.680 

46  32 

6.20 

327.360 

3  32  52 

.83 

96.624 

1  02  54 

3.15 

166.320 

48  15 

6.30 

842.640 

3  36  18 

.84 

97.152 

1  03  15 

3  20 

168.960 

49  58 

6.40 

337.920 

3  3943 

.85 

97.680 

1  03  35 

3.25 

171.600 

51  41 

6.50 

343.200 

3  43  08 

.86 

98.208 

1  03  56 

3.30 

174  240 

53  24 

3.60 

348.480 

3  46  34 

87 

98.736 

1  04  17 

3.35 

176.880 

55  07 

6.70 

353.760 

3  49  59 

.88 

99.264 

1  04  37 

3.40 

179.520 

56  50 

6  80 

359.040 

3  53  24 

.89 

99.792 

1  04  58 

3.46 

182.160 

58  33 

6.90 

364.320 

3  56  50 

.90 

100.320 

1  05  19 

3.50 

184.800 

2  00  16 

7.00 

369.600 

4  00  15 

When  the  running  is  tolerably  easy,  instead  of  taking  a 
series  of  short  courses,  it  is  often  better  to  insert  a  curve  at 
once,  selecting  one  which  is  likely — as  near  as  can  be  guessed 
— to  coincide  with  the  probable  final  location;  for  in  this  way 
truer  results  can  be  arrived  at  than  by  a  series  of  independent 
courses. 

54.  As  regards  the  Instrument-work  itself,  the  method  of 
reading  angles  as  so  much  "to  the  right  "  or  "  to  the  left"  is 


52  KAILROAD   LOCATION. 

decidedly  feeble.  The  best  way  is  to  start  with  the  verniers 
reading  zero  when  the  telescope  is  pointing  towards  the  mag- 
netic, or  still  better,  the  true  north;  then  the  first  angle  read 
is  the  magnetic  (or  true)  bearing  of  the  first  course.  On  mov- 
ing the  instrument  up  to  the  front  picket,  the  horizontal  circle 
should  be  kept  clamped,  and  the  reading  of  the  vernier  again, 
when  the  instrument  is  next  set  up,  constitutes  a  check  on  the 
former  reading;  for  though  there  will  probably  have  been 
some  slipping  of  the  plates,  owing  to  the  shaking  while  being 
carried  from  one  station  to  the  other,  an  error  of  a  degree  or  so 
is  easily  detected.  When  the  telescope  is  pointed  to  the  back- 
sight the  verniers  should  then  read  the  same  as  they  did  at  the 
other  end  of  the  line,  and  thus  for  the  next  course,  on  turning 
through  the  required  angle,  it  will  be  its  bearing— magnetic 
or  true  as  the  case  may  be — that  is  read.  The  compass-reading 
should  also  be  taken  for  each  course,  at  each  end  of  the  course, 
which  thus  forms  an  additional  check  on  the  work,  and  also 
detects  local  attraction.  For  if,  when  the  instrument  is  set  up, 
the  needle  does  not  on  any  course  read  the  bearing  correspond- 
ing with  the  vernier  reading  (if  the  zero  corresponds  with 
the  magnetic  north)  or  does  not  give  the  difference  in  the 
readings  equal  to  the  "  variation,"  if  the  zero  corresponds 
with  the  true  north,  if  the  work  is  correct,  the  cause  is  either 
the  change  in  variation,  or  local  attraction,  or  both  these 
causes  combined.  If  the  instrument  is  a  good  one  there  is  no 
need  to  read  by  more  than  one  vernier.  (See  Sec.  45.)  But  it 
should  usually  be  tlie  same  vernier  that  is  read,  and  that 
vernier  will  then  always  be  on  the  same  side  of  the  transit-line. 
If,  however,  the  line  of  collimation,  from  some  cause  or  other, 
such  as  a  defective  object-slide  which  cannot  be  remedied  in 
the  field,  is  unreliable,  the  error  can  be  counteracted  to  a  large 
extent  by  taking  the  bearings  with  the  same  vernier  on  op- 
posite sides  of  the  line  at  alternate  stations. 

55.  With  the  bearings  taken  as  above,  or  in  fact  taken  in 
any  way,  the  most  satisfactory  method  of  plotting  the  work  is 
by  means  of  LATITUDES  AND  DEPARTURES.  This 
method  involves  a  little  extra  work,  but  its  advantages  over 
the  ordinary  protractor  method — or  even  the  method  of 
"chords"  or  "natural  tangents" — are  so  great  as  to  make  the 
few  minutes  extra  time  taken  in  preparing  the  notes  time  well 
spent.  The  main  advantage  of  this  method  is  that  an  error 


RAILROAD   LOCATION. 


53 


made  in  plotting  one  station  is  not  transmitted  to  the  next,  as 
in  the  ordinary  methods,  for  each  station  is  plotted  entirely 
independent  of  the  previous  one  ;  and  thus  of  course  we  can 
plot  any  one  part  of  the  location  on  the  plan  in  its  right  posi- 
tion, without  having  to  work  through  from  the  beginning. 
Again,  if  we  know  the  position  of  the  point  we  are  making 
for,  we  can,  without  keeping  a  continuous  plot  of  the  work, 
tell  at  any  station  how  much  we  are  off  our  direct  route,  and 
what  course  we  ought  to  steer  to  strike  the  point  we  are  mak- 
ing for.  The  method  of  keeping  and  plotting  the  notes  is  best 
shown  as  follows: 


1600. 

987.2 

518. 
164.5 


44.00 


FIG.  9. 


Suppose  Fig.  9  represents  the  first  five  courses  of  a  prelimi- 
nary line,  the  notes  for  these  courses  will  then  be  kept  thus: 


Sta. 

Dist. 

Read. 

Bearing. 

Lat. 

Dep. 

Total 
Lat. 

Total 
Dep. 

0 

1036 

60° 

N.  60°  E. 

518 

897.2 

10.36 

1064 

90° 

E. 

0 

1064. 

518. 

897.2 

21.00 

550 

130° 

S.  50°  E. 

-8W.5 

421.3 

518. 

1901.2 

26.50 

950 

30° 

N.  30°  E. 

822.7 

475. 

164.5 

2382.5 

36.00 

800 

-40° 

N.  40°  W. 

613.8 

-514.2 

987.2 

2857.5 

44.00 

1600. 

2343.3 

Readings  which  give  a  westerly  course  should  be  considered 
negative;  so  also  should  latitudes  south  and  departures  west, 
as  shown  above.  Then 


54  RAILROAD   LOCATION". 

Latitude  for  any  Sta.  —  Distance  X  Cosine  of  Bearing, 

Departure    "        "     =        "        X  Sine 
and 

Total  Latitude  for  any  Sta.  = 
Total  Latitude  for  preceding  Sta.  -{-  Lat.  for  preceding  Sta. 

Total  Departure  for  any  Sta.  = 

Total  Departure  for  preceding  Sta.  -f-  Dep.  for  preceding  Sta. 
The  term  "Latitude"  is  an  abbreviation  of  "  Diff ereuce  of 
Latitude."  The  terms  "  Cosines"  and  "  Sines"  are  more  ap- 
propriate when  the  bearings  are  kept  with  no  particular  refer- 
ence to  the  true  or  magnetic  meridian. 

By  the  aid  of  cross-section  paper  (if  true  to  scale)  we  can 
plot  the  survey  from  the  notes  with  only  a  straight-edge. 
Thus,  e.g.,  to  find  the  position  of  Sta.  26  +  50,  we  read  oif 
along  the  N.  and  S.  base  a  distance  to  the  north,  equivalent  to 
164.5  feet,  and  along  the  E.  and  W.  base  a  distance  to  the 
east  equivalent  to  2382.5  feet;  the  intersection  of  the  co- 
ordinates from  these  two  points  gives  the  position  required. 

On  a  long  plan,  if  we  have  the  base-lines  drawn  straight, 
and  points  accurately  scaled  off  along  them  at,  say,  every  1000 
feet,  there  is  very  little  chance  of  making  an  appreciable  error 
in  the  plotting  of  the  plan  if  the  notes  are  correctly  worked 
out.  But  although  this  method  is  undoubtedly  the  best, 
unless  the  notes  are  well  checked,  it  is  very  liable  to  give  rise 
to  errors  owing  to  arithmetical  mistakes  in  the  notes  them- 
selves. But  where  good  work  is  wanted,  and  in  cases  where 
probably  the  method  of  plotting  by  "chords"  or  "natural 
tangents"  would  otherwise  have  been  used,  the  method  of 
Latitudes  and  Departures,  well  checked,  gives  far  better 
results,  and  probably  takes  no  longer  than  the  other  ways. 

56.  The  only  way  in  which  to  feel  sure  that  there  are  no 
appreciable  mistakes  in  the  transit-work  is  to  check  the  bear- 
ing of  the  alignment  every  now  and  again  by  an  observation 
for  azimuth.  This  should  be  done,  if  possible,  before  starting 
the  survey,  or  in  any  case  as  soon  after  as  possible,  and  the 
notes  then  already  taken  reduced  to  their  true  bearings.  By 
taking  the  magnetic  pole  as  the  standard  of  our  bearings,  we 
have  no  means  of  applying  an  accurate  check  to  the  work  at  a 
later  period;  but  if  we  start  with  the  vernier  at  zero,  when 
the  telescope  is  pointing  to  the  true  north,  we  can  then  check 
our  course  at  any  time  on  the  survey. 


RAILROAD  LOCATION.  55 

Engineers  generally  fight  rather  shy  of  anything  in  connec- 
tion with  astronomical  work;  but  considering  that  it  is  almost 
as  easy  to  check  the  alignment  by  means  of  a  star  as  by  any 
known  point  on  the  Earth's  surface,— and  usually  much  more 
accurate, — it  is  a  great  pity  that  observations  for  azimuth  are 
not  used  more  frequently  than  they  are.  It  is  so  much  more 
satisfactory  for  the  transit-man  himself  to  know  if  he  is  doing 
good  work;  and  considering  that  the  transit-line  is  usually 
taken  as  the  basis  of  all  the  plans  to  be  afterwards  constructed, 
every  possible  means  of  checking  the  work  should  be  used. 

57.  The  handiest  methods  of  obtaining  the  true  north  are 
the  following,  one  of  which  is  applicable  in  most  northern 
latitudes  about  every  6  hours,  and  can  be  applied  without  any 
knowledge  at  all  of  astronomical  work: 
A.  By  a  Maximum  Elongation. — In  Fig.  10  let 

Z  represent  the  zenith, 

P       "        the  pole, 

8  "  the  Pole-star  (Polaris). 
Then  the  small  circle  round  the  pole 
shows  the  path  and  direction  of  the  star's 
motion,  the  time  taken  in  making  the 
circuit  being  nearly  24  hours.  Now  the 
radius  of  this  small  circle  in  angular 
measure  is  only  about  equal  to  1£°  (or2| 
diameters  of  the  sun),  so  that  the  apparent 
motion  of  the  pole-star  in  azimuth  (i.e., 
horizontally)  will,  when  due  east  or  west, 
be  nothing  at  all,  and  for  several  minutes 
together  when  about  east  or  west  the 
motion  will  be  inappreciable  to  ordinary  railroad  transits. 
Thus  if  we  know  about  what  time  the  star  will  be  at  its  east 
or  west  elongation, — i.e.,  due  east  or  due  -west, — and  also  the 
amount  in  azimuth  by  which  when  at  those  points  it  will  be 
distant  from  the  pole,  we  can,  by  setting  the  telescope  on  the 
star  when  at  either  of  its  elongation  sand  apply  ing  the  required 
correction  in  azimuth,  obtain  the  direction  of  the  true  north. 
The  following  table  shows  approximately  the  times  at  which 
the  elongations  will  occur.  The  amount  of  the  correction  in 
azimuth,  which  really  equals  the  angle  WZP  (or  EZP),  may 
be  found  by  solving  the  spherical  right-angled  triangle  WPZ, 
the  angle  at  W  being  90°,  the  side  &  being  equal  to  90° — the 


56 


RAILROAD  LOCATION". 


"  declination"  of  the  star.  For  Declinations  of  Stars  see  Table 
in  Sec.  213.  Thus  we  have 

Sin  azimuth  —  cos  (dec.)  sec  (lat.), 

PZ  being  the  complement  of  the  latitude  of  the  place  of  ob- 
servation. Thus  suppose  in  latitude  50°  N.,  in  January  1889,  we 
have  the  telescope  clamped  on  Polaris  at  its  eastern  elongation, 
the  vernier  reading  2°. 05';  then  the  sine  of  the  azimuth  correc- 
tion =  .0349,  which  gives  a  value  for  the  correction  of  2°.  00, 
so  that  the  telescope  will  be  pointing  due  north  when  the 
vernier  is  set  to  read  0°.05'.  (See  note  D,  Appendix.} 

TIMES  OF  ELONGATIONS  OF  POLARIS. 


1st  Day. 

llth  Day. 

21st  Day. 

Month. 

Eastern. 

Western. 

Eastern. 

Western. 

Eastern. 

Western. 

h  m. 

h.  m. 

h.  m. 

h.  m. 

h.  m. 

h.  m. 

Jan  

0.39  P.M. 

0.31  A.M. 

11.59A.M. 

11.47P.M. 

11.20A.M. 

ll.OSp.M. 

Feb.... 

10.36  A.M. 

10.25P.M. 

9.57 

9.45 

9.18 

9.06    ' 

Mar.  .  .  . 

8.46    " 

8.34 

8.07 

7.55 

7.27 

7.16    ' 

April.  .  . 

6.44    " 

6.32 

6  05 

5.53 

5.26 

5.14    ' 

May.  .  .  . 

4.46    " 

4.34 

4.07 

3.55 

3.28 

3.16    ' 

June.  .  . 

2.45    " 

2.33 

2.05 

1.54 

1.26 

1.14    ' 

July.... 

0.47    " 

0.35 

0.04 

11.  56  A 

M. 

11.25  p 

M. 

11.  17  A. 

M. 

"    * 
Aug  .  .  . 

10.42  P.M. 

10.  34  A 

M. 

10.03P 

f. 

9.55 

9.23 

9  15    k 

Sep..... 

8.40    " 

8.32 

8.01 

7.53 

7.21 

7.13    ' 

Oct  

6.42    " 

6.34 

6.03 

5.55 

5.24 

5.16    ' 

Nov.  .  .  . 

4.40    " 

4.33 

4.01 

3.53    " 

3.22 

3.14    ' 

Dec.  .  .  . 

. 

2.42   " 

2.34 

2.03 

1.55    " 

1.24 

1.16    ' 

I 

Although  the  hour-angles  from  which  the  above  times  are 
2alculated  vary  year  by  year  and  in  different  latitudes,  they 
may  be  considered  to  be  sufficiently  correct  between  the  years 
1890  and  1900,  and  between  latitudes  25°  and  65°  N.  Where 
extreme  accuracy  is  wanted,  the  time  of  observation  may  be 
calculated  as  in  note  D,  Appendix.  The  above  times  increase 
by  about  4  minutes  every  10  years.  But  as  these  elongations 
occur  only  at  intervals  of  12  hours,  more  or  less,  it  is  well  to 
have  some  other  means  of  obtaining  the  true  north,  which 
can  be  used  when  the  above  method  is  inapplicable.  The  two 
following  are  similar  to  one  another  in  principle,  but  occur 
about  12  hours  apart,  and  from  5  to  7  hours  from  the  time  of 
the  elongations  given  above. 

B.  In  Fig.  11  let  P  be  the  pole  and  8  the  Pole-star,  and  let  A 


RAILROAD   LOCATIOK.  57 

represent  Alioth  (e  Ursae  Majoris),  and  C  represent  the  star 
"  Gamma"  (y)  Cassiopeia.  The  arrows  and  dotted  lines  show 
the  paths  and  the  directions  of  the  •  ,,— <^ 

motion  of  the  three  stars.     The 
positions  of  the  stars  in  the  figure 
are  those  which  they  would  oc-       * 
cupy  about  the  time  of  the  western    \.  ( 
elongation  of  Polaris;  but  since  the  "*-• 
complete  circuit  occupies  about 
24  hours,  we  see  that  in  about  6 
hours  C  will  be  about  vertically 
under  S.   When  this  occurs  (i.e., 
when  S  and   0  are  in  the  same  ~~ 

vertical  plane),  clamp  the  tele- 
scope on  Polaris,  and  wait  through  an  interval  of  time  which 
is  to  be  found  from  the  interval  of  29  minutes  30  seconds  for 
Jan.  1,  1889,  by  applying  for  any  later  date  an  annual  correc- 
tion of  -|-  19  seconds.  After  the  lapse  of  this  interval  Polaris 
will  be  due  north. 

C.  The  tliird  method  consists  in  making  use  of  Alioth  in  a 
similar  manner  to  that  in  which  we  have  just  made  use  of  y 
Cassiopeia.  But  in  this  case,  when  Alioth  is  vertically  below 
Polaris,  Polaris  will  be  nearly  at  its  upper  "  culmination" 
(or  "transit,"  as  its  passage  across  the  meridian  is  called),  but 
this  makes  no  difference  in  the  mode  of  procedure.  The  inter- 
val to  wait  when  using  Alioth  was,  on  Jan.  1,  1889,  about  27 
minutes,  and  increases  annually  by  17  seconds.  To  calculate 
the  above  intervals,  see  note  E,  Appendix;  but  for  ordinary 
work  the  figures  given  above  are  sufficiently  correct  as  far 
north  as  70°,  and  as  far  south  as  A  or  Care  visible  at  their  lower 
culminations.  The  altitude  at  which  C  or  A  will  be  above  the 
horizon  when  due  north  equals  about 

Latitude  of  the  place'—  30°; 

so  that  observations  B  and  D  cannot  practically  be  used 
farther  south  than  latitude  35°  N.  If,  however,  the  instrument 
has  a  reflecting  eye-piece,  if  either  observation  B  or  C  is 
needed  farther  south  than  these  limits,  A  and  C  can  be  used  at 
their  upper  culminations,  wrhich  will  take  place  near  the  zenith, 
the  intervals  of  time  and  modes  of  procedure  will  be  the  same 
as  for  the  lower  culminations. 
To  obtain  the  azimuth  of  Polaris  at  any  time  see  Sec.  202. 


58  EAILROAD   LOCATION. 

There  can  be  no  difficulty  about  finding  these  stars  if  it  is 
remembered  that  the  altitude  of  the  pole-star  is  about  equal  to 
the  latitude  of  the  place;  that  the  "  pointers"  pp,  Fig.  11,  point 
towards  it;  that  A  and  C  are  each  about  30°  from  the  pole-star; 
C,  A,  and  8  being  all  three  more  or  less  in  a  straight  Hue. 

The  remarks  made  in  Sec.  45 regarding  the  vertical  axis,  etc., 
should  be  carefully  attended  to.  The  times  at  which  observa- 
tions B  and  C  will  occur  can  be  found  near  enough  by  notic- 
ing the  positions  of  the  stars  themselves. 

In  observation  A  the  instrument  should  be  "  reversed"  on  the 
star  at  the  elongation.  In  observations  B  and  C,  where  the 
star's  motion  in  azimuth  is  comparatively  rapid,  observe,  say,  2 
minutes  before  the  star  is  due  north,  and  then  again  2  minutes 
after  its  transit:  the  mean  result  should  then  be  taken.  An 
error  of  about  2  minutes  in  time  in  observations  B  and  C 
causes  an  error  in  azimuth  of  about  1'.  The  verticality  of  the 
two  stars  should  be  also  tested  by  a  reversal  of  the  instrument. 
58.  In  checking  the  line  by  an  azimuth  observation  as 
already  described,  it  must  be  borne  in  mind  that  the  converg- 
ence of  the  meridians  needs  a  very  important  correction  in 
the  bearings  relating  to  other  points  east  or  west  of  the  place 
where  the  observation  is  taken.  This  may  be  best  shown  by 
means  of  Fig.  12. 

Let  ONEF  represent  a  sector  of  the  northern  hemisphere, 
and  let  A  be  the  point  on  the  earth's  surface  at  which  the 
survey  was  started,  a  continuous  "straight"  line  being  run 
•which  had  at  A  a  bearing  due  west.     After 
//i  we  have  traversed  a  difference  of  longitude 
//   i  which   is  represented  by  the  angle  EOF 
(or  the  spherical  angle  N)  and  have  arrived 
at  C,  we  shall  be  considerably  south  of  the 
point  A,  our  line  having  taken  the  course 
AC  in  the  figure:  so  that,  if  at  C  we  take 
an  observation  for  azimuth,  we  shall  find 
our   line   to  have  a  bearing  considerably 
south  of  east;  and   similarly  all  straight 
lines  run  from  A,  either  towards  the  east 
or  west,    have  a  tendency  to  run  to  the 
south;  similarly  in  the  southern  hemisphere 
they  would   have  a  tendency  to  run  to  the 
north.  Thus  in  order  to  run  a  line  from  A  to  a  point  B,  keep- 


RAILROAD   LOCATION". 


59 


ing  in  the  same  latitude  the  whole  way,  it  becomes  necessary 
to  run  it  as  a  curve.     (See  Sec.  209.) 

Now  the  amount  of  this  increase  in  bearing  from  the  north 
is  equal  to  the  convergence  of  the  meridians  between  the  two 
places,  so  that  in  the  case  of  A  and  B  the  difference  in  the 
bearings  of  the  same  straight  line  obtained  by  observation  at 
each  place  will  be  represented  by  the  angle  BPA,  which  for 
ordinary  work  we  may  consider  equal  to  the  difference  of 
longitude  of  the  two  places  multiplied  by  the  sine  of  their 
mean  latitude.  (See  note  F,  Appendix.)  Thus  if  in  latitude  40° 
north  we  start  a  straight  line  from  A  due  west  and  run  it  to  C 
through  1°  of  longitude,  the  bearing  obtained  by  observation 
at  C' should  be  S.  89°,  21'  W.  But  since.it  often  needs  some 
calculation  to  ascertain  the  difference  of  longitude,  we  can 
best  proceed  in  ordinary  work  by  finding  from  the  following 
table  the  correction  to  be  applied.  Thus  if  in  latitude  50°  N. 
we  have  run  a  line  which  gives  a  total  amount  of  easting  or 
westing  (i.e.,  Total  Departure)  equal  to  60  miles,  the  amount 
of  the  correction  to  apply  will  be 

60  X  1'  02"  =  1°  02'. 

TABLE  OF  CORRECTION  FOR  CONVERGENCE  FOR 
1  MILE  OF  EASTING  OR  WESTING. 


Lat. 

Correction 
for  1  mile. 

Lat. 

Correction 
for  1  mile. 

Lat. 

Correction 
for  1  mile. 

10° 

9".  18 

27° 

26".  52 

44° 

50".  19 

11° 

10".  13 

28° 

27".  66 

45° 

52".  00 

12° 

11".  07 

29° 

28".  85 

46° 

53".  83 

13° 

12"  M 

30° 

30".  03 

47° 

55".  67 

14° 

12".  98 

31° 

31  ".26 

48° 

57".  67 

15° 

13".  96 

3-2° 

32".  49 

49° 

59".  83 

16° 

14".  93 

33° 

33".  83 

50° 

'02".  00 

17° 

15".  92 

34° 

35".  17 

51° 

04".  17 

18° 

16".  91 

35° 

30"  50 

52° 

06".  67 

19° 

17".  93 

36° 

37".  83  * 

53° 

09".  17 

20° 

18".  94 

37° 

39".  17 

54° 

11".  67 

21° 

19".  98 

38° 

40".  67 

55° 

14".  33 

2-2° 

21  ".02 

39° 

42".  17 

56° 

17".  17 

23° 

2-,"'.  10 

40° 

43".  67 

57° 

20".  00 

24° 

23".!  7 

41° 

45".  17 

58° 

23".  00 

25° 

24".  30 

400 

46".  85 

59° 

26".  25 

2G° 

25".  38 

43° 

48".  52 

60° 

30".  00 

This  shows  the  necessity,  when  running  a  long  continuous 
survey,  of  referring  all  bearings  to  an  Initial  Meridian,  either 


60 


EAILROAD   LOCATION. 


at  the  point  from  which  the  survey  started,  or  at  a  point  near 
its  centre.  The  same  remarks  of  course  apply  to  magnetic 
courses  to  a  certain  extent,  but  in  this  latter  case,  on  account 
of  the  constantly  changing  variation,  such  corrections  are 
hardly  practical. 

59.  When  the  transit-line  crosses  a  river  or  ravine  or  some 
other  obstruction  over  which  it  is  difficult  to  obtain  direct 
measurement,  the  best  way  to  proceed  is  by  Triangulating, 
using  whichever  of  the  methods  shown  in  Fig.  13  is  most  ap- 
plicable to  the  case. 


FIG.  13. 


The  angles  at  A  and  F  each  =  90°,  and  at  /,  K,  and  L  = 
60°;  then 

AB  =  ACtsm  C, 
DE=  DFsecD, 


J  K.  —  J  Li  ^^  zLx/, 

where  H=  180°  -  (/+  G). 

If  the  ground  on  which  we  measure  our  base  has  a  tolerably 
uniform  slope  in  the  direction  of  the  base,  it  is  better  to  take 
direct  measurement  along  the  surface  of  the  ground  and  mul- 
tiply the  distance  so  obtained  by  the  cosine  of  the  inclination 
to  obtain  the  horizontal  distance,  than  to  "break-chain." 
Whatever  difference  in  elevation  there  may  be  between  two 
such  points  as  A  and  B,  if  the  base  measurement  is  reduced  to 
the  horizontal,  the  distance  as  calculated  for  ^47?,  from  the 
angles  observed  with  a  transit,  will  also  be  the  horizontal  dis- 


KAILEOAD   LOCATION. 


61 


tance.  If  the  angles  were  observed  with  a  sextant,  of  course 
this  would  not  be  the  case.  (See  Sec.  144.) 

If,  instead  of  encountering  such  obstructions  as  those  given 
above,  an  obstacle  which  we  are  unable  to  see  across  presents 
itself,  such  as  a  huge  detached  rock  on  which  we  cannot  set 
up  the  instrument,  then  perhaps  as  good  a  way  as  any  to  get 
round  it  is  by  offsetting  the  line  so  as  to  run  past  it  on  a  paral- 
lel one,  and  then  on  the  far  side,  by  equal  offsets,  getting  back 
on  to  the  former  line.  If  the  obstacle,  however,  is  too  large 
to  pass  it  well  by  this  means,  we  can  apply  the  equilateral  tri- 
angle JKL  (Fig.  13).  This  latter  method  is  a  good  one  to  use 
whenever  practicable:  there  is  no  calculation  necessary  in 
connection  with  it,  the  angles  used  are  those  most  favorable 
to  exact  work,  and  where  the  obstacle  can  be  seen  over,  a 
check  can  be  applied  by  observing  the  angle  at  K. 

After  having  run  the  line  a  certain  distance  ahead,  repre- 
sented by  the  amount  L,  it  is  often  necessary  to  "back-up" 
and  start  the  line  again  from  the  instrument  so  as  to  strike  a 
point  a  certain  distance  d  on  one  side  of  the  point  where  the 
first  line  struck;  the  correction  G  for  this  may  be  found  thus: 


tan  C  = 


L 


For  more  on  the  subject  of  triangulation  ,  etc.,  see  Part  III. 

60.  The  LEVELLER'S  WORK  on  preliminary  location 
consists  mainly  in  taking  the  elevation  at  every  full  station, 
and  at  any  intermediate  points  where  he  may  consider  it  ad- 
visable to  do  so.  The  best  form  of  keeping  notes  on  such, 
work  is  the  following: 


Sta. 

B.S. 

Int. 

F.S. 

H.I. 

Elevation. 

B.M. 

4.25 

102  35 

195 

4.8 

'     106.60 

101.8 

+50 

7.3 

99.3 

196 

3.28 

5.61 

100.99 

in  which 

Elevation  in  any  line  =  H.I.  -  F.S.  ) 
or         •«          "      "      "     =H.I.-Int.   \ 

and 

H.I.  in  any  line  =  Elev.  -}-  B.S.  in  preceding  line, 


62  RAILROAD   LOCATION. 

The  "Intermediate"  column  is  sometimes  omitted,  but  the 
insertion  of  it  makes  it  easier  to  check  each  page  by  means  of 
the  difference  of  the  sum  of  the  Back-sights  and  Fore-sights. 

To  apply  this  check  between  two  stations,  A  and  B 
for  instance,  which  have  been  used  as  turning-points,  add  to- 
gether all  the  back  sights  between  A  and  B  (including  the 
B.S.  at  A,  but  excluding  it  at  J5);  then  add  together  all  the  fore- 
sights (excluding  the  F.S.  at  A,  but  including  it  at  B):  the 
difference  of  these  two  sums  should  equal  the  difference  in 
elevation  of  A  and  B.  If  the  sum  of  the  back-sights  is  greater 
than  the  sum  of  the  fore-sights,  B  is  higher  than  A;  but  if  less, 
then  lower. 

The  levels  should  be  worked  out  in  the  field  whenever  time 
permits,  for  reference  on  the  work.  The  profile  for  each  day's 
work  should  be  made  out  when  possible  in  the  evening  of  the 
day  on  which  the  work  was  done. 

As  regards  the  precision  of  a  line  of  levels  run  as  above, 
the  probable  error  is  usually  assumed  to  vary  as  the  square  root 
of  the  distance.  The  limit  on  the  British  Ordnance  Survey  is 
0.01  foot  per  mile;  thell.  S.  Coast  Survey  requires  a  limit  of 
0.03  per  mile.  If  we  assume  a  limit  of  0.05  per  mile  for  rough 
work,  the  probable  error  for  any  distance  equals 


0.05  i/inile. 

Thus  in  100  miles  the  probable  error  =  0.50  ft.  For  more  on 
the  subject  of  levelling  see  Parts  II  and  III. 

61.  The  TOPOGRAPHER'S  WORK  consists  principally  in 
taking  the  ground  slopes,  with  more  or  less  accuracy,  at  every 
full  station  and  at  any  intermediate  points  where  he  may  con- 
sider it  necessary,  by  means  of  which  a  contour  plan  may  be 
constructed. 

To  do  this  he  obtains  from  the  leveller  the  elevation  of  each 
station  and  plus  station  at  which  he  has  taken  levels. 

There  is  a  variety  of  methods  in  use  of  obtaining  the  slopes, 
and  the  advantage  of  each  depends  on  the  accuracy  required, 
the  nature  of  the  country,  and  the  vertical  distance  apart  of  the 
contour-lines. 

Where  the  slopes  are  steep  and  accurate  work  is  wanted,  a 
10-foot  slope-rod  with  clinometer  gives  very  good  results,  but 
is  a  cumbersome  sort  of  instrument  to  carry  about. 

Where  5-foot  contours  are  wanted,  a  hand-level  is  very  coil* 


RAILROAD   LOCATION. 


63 


venient,  since  by  considering  the  height  of  the  eye  above  the 
ground  to  be  5  feet,  the  point  corresponding  to  each  contour- 
line  is  located  at  once  by  the  level,— 5  feet  being  an  easy  height 
to  w hich  to  accommodate  ou  e  '8  self , — and  by  pacing  the  distan ce 
between  these  points  we  have  thus  simply  to  enter  the  dis- 
tances in  the  notes  through  which  each  contour  passes.  By 
taking  the  alternate  points  selected  in  this  way,  this  method  is 
of  course  equally  applicable  to  10-foot  contours.  Fig.  14 
shows  how  this  method  is  worked. 


FIG.  14. 

Suppose,  e.g.,  that  for  a  certain  station  the  topographer  ob- 
tains from  the  leveller  the  elevation  of  1823.8,  and  that  he  is 
taking  5-foot  contours.  Then,  if  the  ground  is  as  shown  in 
Fig.  14,  he  proceeds  as  follows:  The  contour-line  nearest  to 
this  elevation  is  that  of  1825  feet,  the  plane  of  which  passes 
about  1  ft.  above  the  ground-level  at  the  station,  so  that  by 
standing  at  the  point  a  he  can  estimate  with-  his  eye  the  amount 
of  1.2  feet,  and  thus  find  the  point  b  which  corresponds  with 
the  contour  of  1825.  Similarly,  standing  at  b  he  finds  c,  and 
so  on  up  the  slope  as  far  as  he  considers  necessary.  Then  re- 
turning to  a,  he  works  in  the  same  way  on  the  lower  side.  If 
the  distances  are  wanted  accurately,  he  should  have  a  man 
with  a  tape  to  assist;  but  as  a  rule,  pacing,  where  it  is  practi- 
cable, gives  good  enough  results.  The  only  notes  to  be  kept  in 
this  case  are  the  distances  out  (right  or  left)  to  the  respective 
contours. 

An  Abuey  hand-level  (with  vertical  arc)  is  also  frequently 
used,  and  gives  good  results.  All  methods,  however,  which 
involve  taking  the  angles  of  the  slopes  themselves  necessitate 
extra  work.  One  method  of  reducing  this  amount  of  labor  is 
to  have  a  set  of  scales  for  the  various  slopes,  each  made  pro- 
portional to  the  cotangent  of  the  inclination;  but  by  the  use 


64 


RAILROAD   LOCATION. 


of  cross-section  paper  and  a  small  protractor  we  can  probably 
do  the  work  equally  well  aiicl  equally  fast. 

The  stadia  method  is  often  found  very  convenient  for  ob- 
taining topography  where  the  above  methods  would  fail  to 
give  good  results. 

But  besides  taking  the  contours,  the  topographer  must  also 
take  note  of  the  courses  of  streams,  etc.,  on  each  side  of  the 
line  within  a  distance  (usually)  of  a  few  hundred  feet.  The 
bearings  of  these  he  can  take  with  a  small  prismatic-compass. 
He  should  also  be  constantly  on  the  lookout  for  anything 
which  may  be  of  service  in  making  up  the  preliminary  esti- 
mates, such  as  indications  of  the  probable  classification,  the 
flood-marks  of  water-courses,  etc.  If  the  topographer  does 
his  work  thoroughly,  he  usually  has  difficulty  in  keeping  up 
with  the  transit  and  level;  but  this  is  rarely  a  disadvantage,  as 
the  chances  are  that  there  will  be  occasional  "backing- up" 
to  be  done  by  the  party  ahead. 

62.  The  GENERAL  PLAN  of  the  "  preliminary  "  survey 
showing  the  alignment,  topography,  etc.,  is  usually  plotted  to 
a  scale  of  400  feet  to  an  inch,  as  in  Figs.  15  and  16,  thus 
agreeing  with  the  horizontal  scale  of  the  profile. 


FIG.  15. 

In  Fig.  15  let  dbcdef  represent  a  portion  of  the  preliminary 
line  as  shown  on  the  general  plan,  plotted  to  a  scale  of  400  feet 
to  the  inch;  and  let  the  line  have  been  run  to  a  -f-  1.25  p.  c. 
grade,  and  the  contours  be  given  for  every  5  feet  vertical. 
Then  if  each  station  at  which  the  instrument  was  set  up  was 
at  "grade,"  the  grade-contour  will  pass  through  each  of  these 
points,  but  gradually  rising  from  one  contour  to  another, 
crossing  them  successively  at  distances  of  about  400  feet  apart;  so 
that  if,  as  in  Fig.  15,  station  a  happens  to  fall  on  a  contour-line, 


RAILKOAD   LOCATION".  65 

the  grade-contour  will  cut  the  next  line  above,  400  feet  farther 
on,  at  c  ;  and  since  the  next  station  d  is  only  200  feet  from  c, 
it  will  be  situated  about  half-way  between  two  of  the  contours. 

Now  this  grade-contour  is  the  line  which,  if  adopted  for  the 
final  location,  would  give  no  cuts  or  fills  at  all,  so  that  it  is  the 
line  which  would  render  the  cost  of  construction  a  minimum. 
The  judgment  of  the  engineer  here  comes  in  to  decide  how 
much  it  is  advisable  to  deviate  from  this  limit.  So  far  the  work 
has  been  more  or  less  mechanical,  for  there  are  usually  enough 
governing  points  along  the  route  to  decide  within  two  or  three 
hundred  feet  the  course  of  the  preliminary  line;  but  fitting  the 
final  location  on  to  the  plan  is  quite  another  matter.  Suppose 
that  the  engineer  considers  that  the  straight  line  AB  (Fig.  15) 
is  about  where  the  final  line  should  be  located.  Then  the 
shaded  portions  in  the  figure  show  cuts  and  fills  alternately — 
shaded  vertically  beiflg  "  cut,"  and  horizontally  "fill  ;"  and 
the  points  where  the  line  AB  intersects  the  grade-contour  will 
of  course  be  the  "  grade-points."  The  amount  of  centre  cut 
and  centre-fill  can  be  read  off  at  any  point — not  by  scaling, 
but  by  counting  the  number  of  contour  spaces  there  are  be- 
tween the  line  AB  and  the  grade-contour.  Thus,  e.g.,  at  a 
point  in  AB  opposite  c,  there  are  2£  contour  spaces,  equivalent 
to  12|  feet  vertical,  so  that  at  this  point  we  should  have  a  12|- 
ft.  centre-cut.  By  taking  in  this  way  a  few  points  here  and 
there,  the  engineer  can,  by  means  of  Table  XIV,  form  a  fair 
idea  of  the  number  of  cubic  yards  in  each  proposed  cut  or  fill, 
making  allowance  of  course  where  the  surface-slope  is  steep, 
as  shown  in  Sec.  69. 

In  this  way,  then,  there  is  no  great  difficulty  in  obtaining  a 
line  which  will  make  the  cuts  balance  the  fills,  this  being  sim- 
ply a  matter  of  a  few  trials.  Where  curvature,  however,  is 
involved,  it  is  not  so  much  the  question  of  balance  as  of  the 
total  amount  of  cut  and  fill,  which  needs -consideration. 

By  having  the  various  curves  drawn  on  a  horn  protractor, 
or  on  a  piece  of  tracing  cloth,  the  result  of  adopting  any  cer- 
tain curve  can  be  seen  at  once  by  sliding  it  up  and  down  over 
the  plan. 

Then,  again,  a  change  of  grade  for  a  short  distance  may  ap- 
pear advisable,  which  necessitates  altering  the  grade-contour. 
The  question  of  overhaul,  too,  has  to  be  considered,  and  the 
avoidance  as  much  as  possible  of  long  shallow  cuts.  The 


66  RAILKOAD   LOCATION. 

probable  classification,  too,  will  of  course  affect  the  balance 
of  cuts  and  fills.  The  advisability  of  raising  the  grade  to  avoid 
an  expensive  rock-cut  also  needs  consideration.  A  little  ex- 
perience, however,  goes  a  long  way,  and  the  engineer  usually 
finds  that  there  is  little  doubt  to  a  few  feet  as  to  where  the  line 
ought  to  go. 

63.  The  main  features  of  the  final  location  having  been  de- 
termined as  above,  and  drawn  on  the  plan,  the  approximate 
position  of  the  points  of  curvature,  etc.,  can  be  taken  off  by 
scale,  and  the  line  thus  located  on  the  ground;  any  little  alter- 
ations being  made,  the  advantages  of  which  have  become 
apparent  when  the  line  is  seen  actually  staked  out. 

A  fresh  set  of  levels  must  of  course  be  taken  over  the  new 
alignment,  and  a  profile  constructed  showing  the  rates  of  grade, 
etc.,  finally  adopted. 

As  regards  compensating  for  curvature  where  transition  curves 
are  not  used,  the  rate  of  grade  should  be  changed  at  the  P.O. 
and  P.T.  Many  engineers,  however,  prefer  making  the  change 
at  the  nearest  "full  "  station;  it  makes  little  difference,  how- 
ever, which  way  is  adopted. 

Bench-marks  should  be  given  at  distances  of  a  third  of  a  mile 
apart  or  so,  and  guard-stakes  set  solidly  beside  the  hubs.  •  If 
the  location  is  being  "rushed,"  there  is  no  need  to  fill  in  the 
transition  curves,  for  that  can  be  done  equally  well  by  the  sec- 
tion-engineer when  he  takes  over  the  work  for  construction. 
When  these  curves  are  omitted,  however,  it  should  be  so  shown 
on  the  plan,  as  in  Fig.  16. 

64.  It  often  happens  that  after  the  line  is  located  a  consider- 
able distance  ahead  an  alteration  in  the  alignment  is  deemed 
advisable,  necessitating  a  shortening  or  lengthening  of  a  cer- 
tain portion  of  the  line.  This  causes  a  break  in  the  "through- 
chainage."     Such  a  break  as  this  should,  wherever  possible, 
be  referred  to  a  point  where  there  is  a  change  of  grade,  or  at 
least  to  a  point  on  a  tangent,  so  as  to  simplify  the  running  of 
the  grades  and  curves  as  much  as  possible.     It  should  be  in- 
dicated conspicuously  in  the  notes  and  on  the  plans  and  pro- 
files in  the  form  of  an  equation;  the  station  on  the  line  which 
comes  first  being  read  first.     Thus  if  the  left-hand  side  of  the 
equation   is  the  greater,    it   means    that  the   line  has  been 
lengthened;  but  if  the  right-hand  side  be  the  greater,  it  has 


RAILROAD   LOCATION.  67 

been  shortened  by  an  amount  equal  to  the  difference  of  the 
two  sides. 

65.  The  method  of  locating  described  above  is  of  course 
suitable  only  to  rolling  or  mountainous  country;  but  where 
there  is  any  doubt  as  to  whether  or  not  it  is  better  to  lake  con- 
tours, the  engineer  may  generally  come  to  the  conclusion  that 
it  is  better  to  do  so.  There  is  among  some  engineers  an  idea 
that  the  time  spent  in  taking  the  topography  might  have  been 


FIG.  16. 

better  used  in  running  a  series  of  trial  lines.  Of  course  in 
many  cases  this  is  true;  but  it  must  be  remembered  that  a  pre- 
liminary line  with  topography  well  taken  to  a  distance  on 
either  side  of,  say,  500  feet  (as  is  perfectly  feasible  in  ordinary 
rolling  country)  covers  a  width  of  1000  feet  so  completely,  as 
to  render  the  running  of  a  trial-line  within  that  area  entirely 
needless;  and  that  in  order  to  settle  the  question  absolutely 
as  to  the  location  through,  say,  a  valley  half  a  mile  wide,  two 
or  at  the  most  three  lines  run  as  above  are  all  that  can  ever  be 
required;  while  by  the  method  of  trial  lines  how  many  are 
needed  before  the  engineer  can  feel  satisfied  that  he  has  finally 
obtained  as  good  a  line  as  can  be  got  ?  And  then  it  is  only  the 
best  of  the  trial-lines  that  is  usually  selected,  which  in  all  prob- 


68  RAILROAD   LOCATION. 

ability  will  be  inferior  to  the  line  selected  from  the  contour 
plan. 

Besides  this,  if  topography  is  taken,  the  engineer  can  at  any 
future  time  show  evidence  as  to  the  advisability  of  having 
adopted  the  route  which  he  finally  selected.  It  is  a  duty  he 
owes  to  himself  as  well  as  to  the  Railway  Company  to  be  able 
to  prove  that  the  location  has  been  good,  and  how  is  he  to  do 
this  if  he  has  simply  trusted  to  the  correctness  of  his  eye  ? 

66.  In  country  where  the  running  is  easy, -one  or  two  trial- 
lines  usually  show  pretty  closely  where  the  final  line  ought  to 
go,  for  the  long  courses  may  then  be  converted  into  tangents, 
and  curves  be  substituted  for  the  shorter  ones  as  in  Fig.  17. 


FIG.  17. 

If  the  long  courses  predominate,  it  is  usually  better  to  get 
their  location  fixed  first,  and  then  join  them  by  curves;  but 
when  the  shorter  ones  are  in  excess,  it  is  the  curves  that  have 
to  be  first  located,  and  the  tangents  made  subservient  to  them. 

If  the  notes  of  the  courses  are  kept  by  ' '  Latitudes  and  De- 
partures," the  exact  curve  necessary  to  replace  such  courses 
as  ABC  can  be  at  once  found  according  to  Sec.  77. 

67.  An  engineer  with  a  good  "  eye"  can  often  tell  by  mere- 
ly looking  over  the  ground  what  degree  of  curve  is  wanted  to 
fit  the  surface,  i.e.,  where  the  difference  between  a  3°  30'  and 
a  3°  45'  makes  very  little  difference.  Table  II,  of  Tangents 
and  Externals,  is  a  good  guide  to  this  in  many  cases.  For 
instance,  by  getting  into  position  near  the  apex  of  the  re- 
quired curve,  the  engineer,  with  the  aid  of  a  hand-level  and 
a  prismatic  compass,  can  often  tell  about  how  far  from  where 
he  is  standing  the  curve  should  pass.  Thus,  suppose  he 
finds  the  angle  of  intersection  to  be  about  40°,  and  that  the 
curve  should  pass  about  120  feet  from  the  apex:  he  then 
finds  from  the  Table  that  for  an  intersection-angle  of  40°  a  1° 
curve  gives  an  external  distance  of  368  feet,  therefore  tlie 


RAILKOAD  LOCATION.  69 

degree  of  curve  which  he  wants  will  be  found  by  dividing  this 
by  120;  thus  a  3°  04'  curve  will  probably  suit  the  case. 

Where  the  APEX  of  a  curve  can  be  located  without  much 
trouble  it  is  always  better  to  do  so;  and  of  course  this  applies 
more  especially  to  places  where  extreme  accuracy  in  the  centre- 
line is  of  importance;  such  as  where  bridge-work  or  trestliug 
are  required  in  the  neighborhood  of  the  curve. 

68.  The  balancing  of  cuts  and  fills  in  comparatively  level 
country  is  usually  unadvisable,  partly  on  account  of  the  extra 
expense  involved  by  the  matter  of  over-haul,  but  mainly  be- 
cause, though  the  dump  should  be  kept  as  high  as  possible, 
cuts  in  such  country,  and  especially  long  shallow  ones,  gener- 
ally add  very  considerably  to  the  operating  expenses.     Thus 
the  amount  of  borrow  in  such  cases  may  often  with  economy 
be  made  very  considerable. 

69.  On  work  of  this  sort  the  line  is  generally  located  first, 
and  then  the  grades  fixed  by  means  of  the  profile.     This  is 
usually  done  by  straining  a  piece  of  silk  along  the  surface-line, 
by  means  of  which  the  effect  of  adopting  certain  grades  cor- 
responding with  the  various  positions  of  the  thread  can  at  once 
be  seen;  and,  judging  by  the  depth  of  centre-cut  or  fill,  a  fair 
estimate  can  thus  be  made  of  the  amount  of  excavation  and 
embankment  required. 


Grade  Line               \  / 

\ 1 


FIG.  18. 


Where  the  work,  however,  is  comparatively  heavy,  the  fol- 
lowing method  will  be  found  to  give  considerably  better  results: 
Suppose  the  dotted  surface-line  in  Fig.  18  to  be  part  of  a  pro- 


70  RAILROAD   LOCATION. 

file  on  which  we  want  to  fix  the  grades  so  as  to  make  the  cuts 
and  fills  balance,  and  that  in  this  case  we  wish  to  make  a  por- 
tion of  cut  A  together  with  the  whole  of  cut  B  sufficient  to  fill 
the  hollow  G.  On  a  piece  of  tracing  cloth,  say  10  inches  long, 
draw  a  straight  heavy  line  which  is  to  be  the  grade-line;  then 
turn  to  Table  XIV,  and  see  what  depth  of  cut  is  required  to 
give  1000  cubic  yards  contents  in  a  length  of  100  feet.  Thus 
if  the  cuts  are  to  have  a  20-foot  base  and  slopes  of  14  to  1,  as  in 
Fig.  18,  the  depth  of  cut  required  will  be  about  8.3  feet.  Then 
draw  the  parallel  line  above  the  grade-line  already  drawn  at 
this  distance  from  it,  according  to  the  vertical  scale  of  the  pro- 
file (in  Fig.  18  taken  as  40  feet  to  an  inch);  and  again  above 
that  line  draw  another,  distant  from  the  grade-line  by  an 
amount  corresponding  to  the  depth  of  cut  required  to  give 
2000  cu.  yds.  in  a  length  of  100  feet;  and  then  draw  a  third 
for  3000  cu.  yds.,  and  so  on,  as  many  as  are  required.  Similar- 
ly, on  the  lower  side  of  the  grade-line  draw  lines  as  above, 
suitable  to  the  required  base  and  slopes  of  the  fill.  Place  the 
tracing-cloth  over  the  profile,  as  in  Fig.  18.  If  then  the  hori- 
zontal scale  of  the  profile  is  400  feet  to  an  inch,  take  a  "40" 
scale,  and  scale  oil  along  the  horizontal  dotted  lines  shown  in 
the  figure.  One  division  of  the  scale  then  corresponds  to  100 
cu.  yds.  Thus,  in  order  to  make  the  cuts  balance  the  fills 
(not  allowing  for  shrinkage,  etc.)  the  grade-line  must  be  so 
placed  that  the  sum  of  the  horizontal  dotted  lines  above  it  is 
equal  to  that  of  the  lines  below  it.  By  sliding  the  tracing-cloth 
up  and  down,  a  balance  can  soon  be  obtained.  By  scaling  oif 
and  adding  the  lengths  of  the  lines  together  mentally,  the  con- 
tents of  a  cut  or  fill  can  be  approximated  to  in  a  very  few 
seconds;  or  the  contents  may  be  read  off  by  means  of  the  ver- 
tical divisions  on  the  profile  paper. 

Where  there  is  a  steep  surface-slope,  an  allowance  must  of 
course  be  added  to  the  results  as  obtained  by  the  above  method. 
The  allowance  which  should  be  made  for  this  depends,  com- 
paratively speaking,  very  little  on  either  the  width  of  the  road- 
bed or  the  depth  of  the  cut  or  fill  at  the  centre,  but  depends 
mainly  on  the  slopes  themselves;  so  that  we  may  say  roughly, 
that  the  following  corrections  are  applicable  to  any  ordinary 
depth  of  cut  (or  fill)  or  width  of  road-bed. 

Thus,  if  by  the  above  method  we  make  the  contents  of  a 
certain  cut  to  amount  to  20,000  cu.  yds.,  with  side  slopes  of 


RAILROAD   LOCATION. 


71 


1  to  1,  if  the  average  surface-slope  is  about  10°,  a  fair  estimate 
of  the  contents  will  be  given  by  21,000  cu.  yds. 


Slope  Ratio. 

Surface-slope. 

5° 

10° 

15° 

20° 

1    tol 
Htol 

Ip.c. 
2  p.c. 

5  p.  c. 
8  p.  c. 

8  p.c. 
20  p.  c. 

17  p.  c. 
45  p.  c. 

As  to  the  effect  of  shrinkage,  it  may  generally  be  ignored  in 
dealing  with  the  balancing  of  cuts  and  fills.  (See  Sec.  113.)  A 
simple  rule  in  dealing  with  rock-work  is  to  assume  that  100 
cu.  yds.  of  rock  in  excavation  make  150  cu.  yds.  in  embank- 
ment. 

70.  It  has  been  assumed  so  far  that  in  estimating  the  amount 
of  excavation  and  embankment  the  method  of  centre-heights 
is  used.  In  the  long  run  the  results  so  obtained  may  generally 
be  considered  to  give  sufficiently  close  results  for  most  pre- 
liminary estimates.  But  when  the  surface-slopes  are  such  as 
to  necessitate  continued  corrections  being  applied,  the  average 
slopes  at  the  different  stations  may  be  jotted  down  by  the 
leveller  when  taking  the  elevations  and  the  quantities  worked 
out  according  to  Mr.  Trautwine's  method  of  equivalent  level 
sections,  or  some  similar  process. 


CUKYES. 

71.  Radius.    Degree  and  Length  of  Curve.— Railroad 
curvature  in  Canada  and  the  United  States  is  expressed  in 


terms  of  the  angle  ACS,  Fig.  19,  which  subtends  a  chord,  AS, 


72  RAILROAD   LOCATION. 

100  feet  in  length;  and  this  angle  is  called  the  Degree  of  the 
curve,  and  equals  D.  ^ 

In  curves  of  small  Eatos,  i.e.,  of  large  degree,  D  varies  very 
nearly  inversely  as  the  radius  R. 

To  convert  D  into  R,  we  have  in  the  right-angled  triangle  AEG 


and  to  convert  R  into  D  this  becomes 

,   ......    (2) 


from  which  formula  Table  I  has  been  calculated. 
From  Equations  1  and  2  we  see  that  R  varies  inversely  as 

sin  —  ,   and  since  it  is  only  when  —  is  very  small  that  its 
2  2 

sine  may  be  considered  to  vary  as  the  angle  itself,  it  follows 
that  although  we  may  say  that  the  radius  of  a  10'  curve  is  one 
tenth  that  of  a  1'  curve,  by  considering  the  radius  of  a  10° 
curve  to  be  one  tenth  that  of  a  1°  curve,  we  should,  on  accu- 
rate work,  be  led  into  an  appreciable  error.  Thus  by  Equation 
2, 

R  of  a  1°  curve  =  5729.65  feet, 
and  -Bof  a  10°  curve  =    573.69  feet, 

instead  of    ........      572.96. 

72.  The  general  practice  of  setting  out  curves  on  railroad 
construction  is  by  means  of  50-foot  Subchords,  assuming 
that  the  angle  subtended  by  any  subchord  at  the  centre  G  is 
proportional  to  its  length.  Suppose,  for  instance,  we  wish  to 
locate  a  10°  curve,  we  see  from  Fig.  19  that  since  AB  —  100 
feet,  if  we  wish  to  substitute  for  it  two  separate  equal  chords 
AF&nd  FB,  they  must  each  exceed  50  feet  in  length,  and  the 
length  of  each  must  equal 

AE  cosec  AFG. 

Now  AFC  =  90°-  -  and  AE  =  50;  therefore 
4 

Corrected  50-ft.  chord  =  50  sec  ?.        ...     (3) 

4 


KAILROAD   LOCATION. 


73 


Thus,  instead  of  using  50-foot  chords  it  is  the  lengths  given 
in  the  following  table  which  must  be  used  in  order  that  two  of 
them  may  give  the  same  curve  for  the  same  deflection-angle 
as  would  be  given  by  a  100  foot  chord: 

VALUES  OF  CORRECTED   50-FOOT   CHORDS. 


Deg. 

Chord. 

Deg. 

Chord. 

Deg. 

Chord. 

Deg. 

Chord. 

1° 

50.000 

6° 

50.017 

11° 

50.057 

16° 

50.122 

2° 

50.001 

50.024 

12° 

50.068 

17° 

50.138 

3° 

50.004 

8° 

50.031 

13° 

50.080 

18° 

50.155 

4° 

50.007 

9° 

50.039 

14° 

50.093 

19° 

50.172 

5° 

50.012 

10° 

50.048 

15° 

50.107 

20° 

50.191 

If  the  above  corrections  are  not  applied,  the  curve  that  is 
set  out,  instead  of  passing  through  the  point  ^"will  pass  through 
a  at  a  distance  from  F  =  50  feet,  and  its  radius  r  will  equal 
cF  instead  of  OF,  and 


therefore 


cF=  25  sec  CFA; 


D 

r  =  25  cosec  — 
4 


(4) 


If  we  compare  this  equation  with  Equation  2,  we  see  that 
the  radius  of  a  curve  of  any  given  value  of  D  set  out  by  50-foot 
chords,  according  to  the  usual  method,  is  exactly  equal  to  half 

the  radius  of  a  curve  whose  degree  =  -_  -set  out  by  hundred- 

10 

foot  chords.  Thus  the  radius  of  a  so  called  10°  curve,  if  set 
out  by  50-foot  chords,  actually  equals  one  half  the  radius  of  a 
5°  curve,  i.e.,  573.14  feet,  not  573.69  as  intended. 

To  find  the  corrected  length  of  any  other  subchord,  see  Sec. 
76. 

The  corrections  which  we  have  just  seen  to  be  necessary  to 
accurate  work,  practically  in  a  distance  of  100  feet  amount  to 
nothing  at  all,  but  often  in  the  total  length  of  a  curve  they 
mount  up  considerably. 

For  instance,  a  10°  curve  run  in  on  location  with  a  100-foot 
chain,  which  should  then  of  course  be  a  true  10°  curve,  can- 
not be  expected  to  ' '  come  out"  well  when  tried  on  construc- 
tion with  50-foot  chords;  for  if  the  curve  is  900  feet  long  and 


74  RAILROAD   LOCATION". 

the  instrument  work  and  measurement  absolutely  correct,  it 
will  not  close  by  0.8  foot. 

73.  The  length  of  a  curve,  in  terms  of  100-foot  stations, 
as  measured  along  100-foot  chords,  may  be  at  once  found  by 
dividing  the  total  angle  (C)  at  the  centre,  in  degrees,  by  the 
degree  of  the  curve.  Thus  if  L  =  true  length  of  curve, 


=  ~=      (nearly),       .....      (5) 


where  I  =  angle  of  intersection.  (See  Eq.  7.)  So  that  if  the 
angle  subtended  at  the  centre  of  a  10°  curve  =  40°,  the  length 
of  the  curve  along  the  chords  =  400  feet;  and  this  method,  on 
account  of  its  simplicity,  is  that  usually  adopted  on  railroad 
work  for  the  measurement  of  curves.  But  the  true  length  of 
the  curve  will  of  course  be  greater  than  this  in  the  same  ratio 
as  the  arc  AFB  in  Fig.  19  exceeds  the  100-foot  chord  AB. 
Now  the  angle  at  the  centre  of  a  circle  which  is  subtended  by 
an  arc  equal  to  the  radius  equals 

180° 

—  =  57°.29578, 
it 

so  that  the  true  length  of  a  curve  is  given  by  the  equation 

L  =  57^958  =  57^958 (6) 


Thus  if  C  f=  40°  and  E  =  573.686  feet  (i.e.,  a  10°  curve), 
L  =  400.507  feet, — not  400  feet,  as  in  the  example  above.  Had 
this  10°  curve  been  set  out  with  corrected  50-foot  chords,  it 
would  have  measured  (along  the  chords)  400.38  feet. 

Table  IV  gives  the  length  of  arcs  of  various  curves  sub- 
tended by  100-foot  chords,  from  which  the  true  length  of  a 
curve  may  be  at  once  found. 

74,  Before  proceeding  to  the  more  practical  problems  in 
connection  with  the  setting  out  of  curves  in  the  field,  it  will  be 
well  to  consider  a  few  of  the  more  important  equations  which 
form  the  groundwork  on  which  these  problems  are  built  up. 


RAILROAD    LOCATION.  ?5 

First,  as  regards  the  nomenclature  of  the  various  parts,  as 
shown  in  Fig.  20. 


p.c 


P.T, 


FIG.  20. 


P.C.  =  Point  of  Curve. 

=  Beginning  of  Curve. 
P.T.  =  Point  of  Tangent. 

=  End  of  Curve. 
A  =  Apex. 

/  =  Intersection-angle. 
C  =  Central  angle. 
L  =  Length  of  Curve. 


D  —  Degree  of  Curve,  if  bd 

=  100  feet. 
T  =  Sub-tangent. 
E  =  External  distance. 
M  =  Mid-ordinate  to  Long 

Chord. 

T  =  Long  Chord. 
R  =  Radius. 


These  symbols  will  be  maintained  throughout  this  article 
on  curves. 

75.  Now  because  Aa  and  Ab  are  tangents  to  the  curve  at  a 
and  b,  therefore  OaA  and  ObA  must  each  equal  90°,  and  the 
angle  a  Ab  at  the  apex  must  equal  180°  —  C ;  therefore 


(7) 


Again,  in  the  triangle  bOd,  since  the  angle  at  b  =  90°  —  — , 


therefore  the 


D 


Tangential  Deflect.-Angle  for  a  100-foot  chord  =  -.      .     (8) 

a 


76  KAILEOAD   LOCATION. 

In  the  right-angled  triangle  AOa 

n 

T  = 


2' 
therefore,  by  Equation  7, 

T=R tan  -.....,  (9) 

m 

And  if  in  this  we  substitute  the  value  for  R  given  in  Equa- 
tion 2,  this  becomes 

T  =  50  tan  —  cosec  - (10) 

Again, 

E  =  R  exsec  AOa  ; 

therefore,  by  Equation  7, 

E  =  R  exsec  -. (11) 

And  by  combining  Equations  9  and  11  we  obtain 

E  =  T  cot  -exsec/; 
2  2 

therefore 

E  =  Tt&u-2 (12) 

So  also 

M  =  R  vers  ^-; 

therefore,  by  Equation  7, 


. (13) 

And  by  combining  Equations  11  and  13,  we  obtain 


exsec  5- 

therefore 

Jf=  j£cos- '.    .    .    .    (14) 


RAILROAD    LOCATION. 


Again,  by  trigonometry, 


therefore 


—  =  T  cos  Aab ; 

a 


T= 


(15) 


And  combining  this  with  Equation  9,  we  obtain 
T  =  2R  tan  -  cos  -; 


therefore 


Y=2R  sin  £ 
& 


(16) 


Again,  by  combining  Equations  13  and  16,  we  obtain 
2M  1 


Y= 


vers  - 


therefore 


(17) 


The  above  equations  can  readily  be  followed  by  referring  to 
Sees.  231  and  232. 

The  following  table  may  be  of  assistance  in  selecting  quick- 
ly the  equations  required.  Thus,  suppose  we  have  T  and  Y 
given,  aud  want  R  ;  we  see  at  once  that  Equation  15  will  give 
us/;  and  then,  by  Equation  9,  we  can  obtain  R. 


Given. 

Required. 

Use 

Eq. 

Given. 

Required. 

Use 
Eq, 

E 

D 

1 

R.  Y 

7 

16 

7,  L 

D 

5 

M,Y 

7 

17 

7,   T 

D 

10 

D,I 

L 

5 

D 

R 

2 

/,   R. 

T 

9 

7,    T 

R 

9 

/,  D 

T 

10 

7,  E 

R 

11 

1,    E 

T 

12 

7,  M 

R 

13 

I,    Y 

T 

15 

7,   Y 

R 

'16 

I,   R 

E 

11 

D,L 

I 

5 

I,   T 

E 

12 

R,  T, 

1 

9 

/,   M 

E 

14 

D,  T 

1 

10 

1,   R 

M 

13 

R,E 

I 

11 

/,  E 

M 

14 

T,  E 

1 

12 

/.    F 

M 

17 

R,M 

1 

13 

7,    T 

Y 

15 

E,  M 

I 

14 

7,   R 

Y 

16 

T,  Y 

I 

15 

7,   M 

Y 

17 

78  KAILROAD   LOCATION-. 

PROBLEMS  IN  SIMPLE  CURVES. 
76.  To  lay  out  a  curve  by  deflection-angles.  —  In  Fig. 
20  we  have  already  seen  (Eq.  8)  that  the  angle  Abd  =  _  ;  but 

a 

suppose  we  measure  off  another  100-foot  chord  de  :  then  dbe 

also  =  —   (since    boe  =  2D,  which   makes    Obe  =  90°  —  D). 

2 

Similarly,  we  might  show  that  for  any  number  of  consecutive 
100-foot  chords  the  total  detlection-angle  would,  for  each  one, 

increase  by  the  amount  —  . 

But  though  the  Total  Deflection  -angle  from  the  tangent  is 
proportional  to  the  number  of  full  stations  when  these  are  the 
only  points  given  on  the  curve,  as  we  have  already  seen  in  the 
case  of-  50-foot  subchords,  if  we  insert  intermediate  stations 
without  correcting  the  lengths  of  the  subchords,  the  degree 
of  the  curve  increases  at  once. 

In  order  to  find  the  corrected  length  of  any  subchord 
we  may  proceed  thus:  In  Fig.  21  let  ab  represent  a  hundred- 
foot  chord,  then  the  angle  ale  =  D;  and  let  I  represent  any 
subdivision  of  it  corresponding  with  the.  length,  of  any  uncor- 
rected  subchord:  then  the  corrected  length  Y  will  be  given  by 
Equation  16,  when 

C  :  D  =  l  :  100. 

If  we  then  insert  this  value  of  G  in 
Equation  16,  we  obtain 


,     .     . 

T  being  the  corrected  length  of  the 
nominal  subchord  I.  In  ordinary  work, 
except  where  a  sharp  curve  is  run  con- 
tinuously throughout  with  subchords, 
we  may  ignore  this  correction. 
Not  taking  the  correction  into  account,  the  deflection  for 

any  subchord  is  to  —  as  the  length   of   the  subchord  is   to 
a 

100  feet;  so  that  for  any  subchord  we  have 
minutes  ln  }  =  °'3-Z>  X  LenSth  of  Subchord  in  feet;      (19) 


RAILROAD   LOCATION". 


79 


and  this  equation  applies  to  a  corrected  subchord  if  we  insert 
in  it  its  uncorrected  length. 

Thus  for  a  14-foot  subchord  on  a  3°  curve  the  deflection- 
angle  is  0°  12'.6.  • 

Let  us  suppose  that  we  are  given  a  3°  Curve  to  the  Right  to 
locate  from  a  P.O.  at  Sta.  421  -f  36,  /being  equal  to  12°  30'. 

The  length  of  the  curve  we  find  from  Equation  5 — since  this 
is  assumed  as  the  standard  method  of  measurement  for  rail- 
road curves — to  be  416.7  feet,  therefore  the  P.T.  will  be  at 
Sta.  425  -)-  52.7;  then  if  we  intend  to  use  50-foot  subchords, 
our  notes  will  be  arranged  as  follows: 

3°  CURVE   TO  THE   RIGHT. 

P.O.  =  Sta.  421+36.0. 

P.T.  =  Sta.  425  -\-  52.7. 
Length  of  curve     =  416.7  feet. 
Intersection-angle  =  12°  30'. 
Subtangent  =  209.2  feet. 


Station. 

Distance. 

Deflection. 

Index. 

Remarks. 

421  +  36 

0°0' 

P.O. 

+  50 

14 

0°  12'.6 

0°  12'.6 

422 

50 

0°  45' 

0°  57'.6 

+  50 

1°  42'.  6 

423 

2°  27'.6 

+  50 

3°  12'.  6 

Hub. 

424 

3°-  57'.6 

+  50 

4°  42'.6 

425 

5°  27'.6 

+  50 

6°  12'.6 

+  52.7 

2.7 

0°  02'.4 

6°  15' 

P.T. 

The  Jrafoz-reading  at  any  station  equals  the  sum  of  the  de- 
flections up  to  that  station;  then  since  the  Index-reading  at  the 
P.T.  is  represented  by  the  angle  Aba  in  Fig.  20,  and  Aba  is 

easily  proved  equal  to  — ,  therefore  the  Index-reading  at  the 

P.T.  must  equal  half  the  intersection-angle,  thereby  giving  a 
check  on  the  calculations. 

Having  the  notes  worked  out  as  above,  set  the  transit  up  at 
the  P.C.  as  in  Fig.  22,  and  setting  the  index  to  zero,  clamp  the 
telescope  on  to  a  back-sight  on  the  tangent  (or  on  to  the  apex 


80  RAILROAD   LOCATION. 

if  it  has  been  put  in);  then  for  any  station  the  vernier  must 
read  the  angle  given  in  the  index-column  for  that  station.  But 
suppose  that  when  we  have  reached  Sta.  423  -}-  50  we  are  un- 
able to  see  any  farther.  Then  set  a  huh  (with  a  tack  in  it)  at 
that  station  and  a  back-sight  at  the  P.O.  Set  up  over  the  hub, 


50 

425 


50 


FIG.  22. 

and  setting  the  vernier  back  to  zero,  clamp  the  telescope  on  to 
the  back-sight  and  turn  off  the  remaining  deflections  by  mak- 
ing the  readings  for  the  respective  stations  the  same  as  those 
given  in  the  Index-column.  Thus: 

(1)  When  pointing  to  any  station,  the  vernier  must  always 
be  set  to  read  the  Index-reading  for  that  station. 

(2)  When  on  the  tangent  at  any  station,  the  vernier  must 
always  be  set  to  read  the  Index-reading  for  tfiat  station. 

By  adhering  to  these  two  rules  all  possibility  of  error  as  re- 
gards the  index-readings  is  avoided,  and  with  the  notes  worked 
out  as  above  we  may  locate  the  curve  equally  well  from  either 
end. 

In  order  to  find  the  bearing  of  the  tangent  at  any  station 
with  reference  to  the  tangent  at  the  P.O.,  we  have  simply  to 
multiply  the  index-reading  at  that  station  by  two.  Thus,  if 
in  the  above  example  the  tangent  at  the  P.O.  lies  north  and 
south,  the  bearing  of  the  curve  at  Sta.  423  -|-  50  will  be  N.  6° 
25'. 2  E. 

Usually  in  locating  railroad  curves  there  is  no  necessity  to 
work  out  the  deflections  closer  than  to  the  nearer  half-minute. 

In  places  where  accurate  measurement  is  difficult  to  obtain, 
a,nd  great  exactness  is  wanted,  as  in  giving  centres  for  piers  in 
the  middle  of  a  river,  we  can  often  do  better  work  by  using 


BAILROAD   LOCATION. 


81 


TWO  Transits,  one  on  either  side  of  the  stream,  and  fixing  the 
points  by  intersection.     (See  Sec.  163.) 
77.  To  locate  a  curve  when  the  apex  is  inaccessible. 


FIG.  23. 

— Suppose,  as  in  Fig.  23,  we  have  been  unable  to  locate  the 
apex  of  a  proposed  curve,  but  have  connected  the  two  tangents 
at  a  and  b  by  the  line  ab. 

Then  in  the  triangle  Aab  we  know  the  distance  ab  and  the 
angles  at  a  and  b;  therefore  we  have          " 

ab  sin  b 
Aa  =  — : — 7-. 
sm  A 

where  A  =  180°  —  (a  +  b).  We  can  then  find  the  position  of 
the  P.O.  For  example,  suppose  Aa  =.320  feet  and  I  =  40°; 
then  if  we  wish  to  connect  the  two  tangents  by  a  5°  curve, 
since  the  distance  from  A  to  the  P.C.  is  given  by  Equation  9 
(or  Table  II)  =  417.2  feet,  therefore  the  P.C.  will  be  situated 
97.2  feet  back  on  the  tangent  from  a, 
We  can  then  locate  the  curve  according  to  Sec.  76. 


FIG.  24. 
But  suppose,  instead  of  running  a  direct  line  ab.  it  is  more 


82  RAILROAD    LOCATION. 

convenient  to  run  a  succession  of  courses  as  in  Fig.  24.     Then4 
if  the  position  of  the  stations  a  and  b  has  been  worked  out  by 
"Lats,  and  Deps."  we  cao  at  once  find  the  angles  at  a  and  b 
and  the  length  ab. 
For  instance,  let 

Tot.  Lat.  of  a  =  1020  N.        Tot.  Dep.  of  a  =    560  E. 
Tot.  Lat.  of  b  =    810  K.        Tot.  Dep.  of  b  =  1430  E. 

Then  the  bearing  of  ab  will  be  given  by  the  angle  at  a  in  the 
triangle  aeb  ;  thus 

1430  -  560 
toQ  g=1080-  810  =4148- 

Therefore  the  bearing  of  ab  =  S.  76°  26'  E.,  and  the  length 
ab  =  (1020  —  810)  sec  a  =?  895.2.  Then  if  the  bearing  of  the 
tangent  at  a  =  N.  80°  E.,  and  of  the  tangent  at  b  =  S.  60°  E., 
we  have  in  the  triangle  Adb,  a  =  23°  34'  and  b  =  16°  26'  from 
which  we  can  find  the  position  of  the  P.O.  as  above. 

If  the  notes  have  not  been  already  worked  out  by  Lats.  and 
Deps.  the  position  of  b  with  reference  to  a  can  be  most  easily 
calculated  by  taking  the  tangent  at  a  as  the  N.  and  S.  base. 

78.  To  locate  a  curve  by  offsets  from  a  tangent.— Let 


FIG.  25. 

ab  be  a  tangent  to  the  curve  at  a.    Now  the  value  of  the  tan- 
gential offset  at  any  station  is 

t  =  E  vers  C. 

But  C=  ND  where  N  =  number  of  Stas.  along  the  curve  to 
t,  therefore 

i  =  .B  vers  JVZ) (20) 


RAILROAD   LOCATION.  83 

Similarly,  the  distance  along  the  tangent  from  a  to  the  offset 
t  equals 

X  =  R  sin  ND  ......     (21) 

Thus,  for  example,  suppose  a  falls  at  Sta.  10  -f-  40,  and  we 
wish  from  this  point  to  set  out  a  10°  curve  by  offsets  from  the 
tangent  at  a;  then  at  Sta.  11 


and  the  distance  along  the  tangent  at  which  this  offset  must 
be  set  off  equals 

X  =E  sin  6°  =59.  95  feet. 

The  values  of  t  at  distances  along  the  curves  from  a,  100 
feet  apart,  are  given  in  Table  III,  calculated  by  Equation  20. 

A  formula  that  often  comes  in  handy  in  the  field  for  com- 
puting tangential  offsets,  and  which  is  usually  true  enough 
when  X  does  not  exceed  150  feet,  is 

<  =  g-  (nearly). 

Tangential  offsets  may  pften  be  made  use  of  when,  on  ac- 
count of  some  obstacle  or  other,  the  method  given  in  Sec.  76 
cannot  be  used.  By  offsetting  the  tangent  itself  occasionally, 
as  in  Fig.  26,  we  can  with  ease  run  a  curve  past  a  succession 
of  obstacles,  and  at  the  same  time  keep"  the  offsets  compara- 
tively short. 


FIG.  26. 


Another  occasion  on  which  this  method  can  be  used  to  ad- 
vantage is  when  the  apex,  P.C.  and  P.T.  are  inaccessible. 

Suppose,  by  way  of  example,  that  we  have  to  locate  a  10° 
curve  in  a  position  such  as  is  represented  in  Fig.  27,  the  angle 


84 


RAILROAD   LOCATION. 


of  intersection  having  been  found  according  to  Sec.  77  to  be, 

say,  00°,  and  the  distance  from 
A  to  some  fixed  accessible  point 
e  to  be  723.7  feet :  then  ae  will 
equal  150  feet.  Suppose  we  are 
able  to  begin  running  in  the 
curve  at  c,  a  point  200  feet  along 
the  curve  from  a:  then  the  offset 
at  c  will  equal  34.6  feet,  at  a  dis- 
tance from  a  along  the  tangent 
of  196.2  feet  or  from  e  =  346.2 
feet ;  and  the  offset  at  d,  300 
Fro.  27.  feet  along  the  curve  from  a, 

equals  76.9  feet  at  a  tangential  distance  of  286.8  feet  from  a, 
or  from  e  =  436.8  feet.  Thus  we  have  two  points  c  and  d  fixed 
on  the  curve,  by  means  of  which  we  can  locate  any  other  part 
of  the  curve  accessible  to  them,  as  shown  in  Sec.  76. 

Or,  suppose  we  have  such  a  case  as  that  shown  in  Fig.  28, 
where  we  have  run  the  curve  ab  round  as  far  as  d,  but  find 
that  the  P.T.  is  inaccessible,  and  yet  wish  to  get  on  to  the  tan- 
gent without  adopting  the  method  given  in  Sec.  77.  A  con- 
venient method  of  doing  this  is  to  locate  the  apex  A,  if  acces- 
sible, by  setting  off  from  e,  the  middle  point  of  the  curve,  the 
external  E,  found  by  Equation  11 ;  then  we  have  one  point  on 
the  tangent  Ab. 


Again,  by  running  on  the  curve  as  far  as  is  possible  to  d,  we 
can  there  set  the  vernier  to  read  the  (Index-reading  for  b  -f- 
Diff.  of  Index  between  d  and  b)  —  90°  and  set  off  df=t, 


RAILROAD   LOCATION. 


85 


found  by  Equation  20  :  thus  if  the  Index-reading  for  d  =  40° 
and  for  b  =  60°,  the  vernier  must  read  60°  -f  20°  -  90°  =  — 10°. 

The  angle  ND  in  Equation  20  equals  of  course  the  angle  dob. 
We  thus  have  a  second  point/  on  the  tangent  Ab,  and  therefore 
we  have  its  direction.  Then  by  Equation  9,  since  Ab  =-  T, 
we  can,  by  triangulation,  find  the  distance  of  A  from  some 
accessible  point  p  on  the  tangent  ;  then  bp  =  Ap  —  T.  Or, 
since  by  Equation  21,  fb  =  R  sin  dob  we  can  triangulate  from  p 
to /instead. 

If  A  is  inaccessible  also,  instead  of  proceeding  as  in  Fig. 
27,  we  might  when  at  d  set  the  vernier  to  the  (Index-reading 
for  b  -f-  Diff.  in  Index  between  dJand  b),  which  will  give  a  line 
dli  parallel  to  the  tangent  at  b.  Thus  the  vernier  must  read 
80°.  We  can  then  set  off  pli  =  df,  and  thus  obtain  two  points 
/and  p  in  the  direction  of  the  tangent  Ab  ;  and  since  we  know 
dh  =fp  by  direct  measurement,  and  fp  by  calculation,  we 
thus  have  the  distance  bp. 

Again,  if  we  have  an  obstacle  on  the  curve  itself,  we  can 
run  a  tangent  from  some  point  on  the  curve  which  will  clear 
it,  and  so  connect  the  curve  at  the  further  side  in  a  similar 
way  to  that  shown  in  Fig.  27  ;  or  we  might  run  a  Long  Chord 
past  it  and  lay  it  off  by  ordinates  as  in  Sec.  80.  / 

79.  To  locate  a  curve  by  offsets  from  the  chords  pro- 
duced.— Let  it  be  required  to  locate  a  10°  curve  an  by  offsets 


FIG.  29. 


from  the  chords  produced,  and  let,  for  example,  the  length  of 
the  curve  =  360  feet.     In  Fig.  29— exaggerated  for  the  sake  of 


86  RAILROAD   LOCATION". 

clearness—  let  ab,  bg,  and  gi  be  100-foot  chords,  then  if  eb  is  in 
the  same  straight  line  as  ab  and  is  equal  to  bg,  the  triangle 
beg  is  similar  to  the  triangle  obg;  therefore 

bg  :  H  =  eg  .  bg. 

So  that,  calling  the  chord  bg  =  c,  and  the  chord  deflection  eg 
=  d,  we  have 


but  this  value  of  d  of  course  only  holds  good  when  the  length 
of  the  preceding  chord  (as  ab)  is  equal  to  c. 

Again,  if  fg  =  $eg,  then  the  triangle  bfg  =  the  triangle  axb, 
therefore  xb  =  \eg.     Therefore,  if  t  =  the  tangential  offset, 


(23) 


a  formula  (already  given  in  the  last  section  in  other  terms) 
which  holds  good  for  any  lengths  of  chord,  provided  the  angle 
at  x  =  90°. 

When  c  =  100  feet,  we  also  have  the  formula 

t  =  100  sin  5 

m 

To  find  a  tangent  to  the  curve  at  any  station,  say  i,  we  have 
only  to  set  off  the  value  of  t  =  kg,  obtained  by  Equation  23, 
at  station  g  ;  Id  will  then  be  the  tangent  at  i. 

In  order  to  locate  the  curve  we  therefore  proceed  as  follows: 

Measure  ab  =  100  feet  ;  b  will  then  be  distant  from  ax,  the 
tangent  produced,  by  an  amount  t  =  8.72  feet.  Set  pickets 
at  a  and  b,  and  range  in  the  point  e,  100  feet  from  b  ;  then  g 
will  be  distant  from  e  by  an  amount  d  =  17.43  feet,  and  from 
b  by  100  feet.  Similarly  we  can  locate  i. 

But  the  60-foot  subchord  in,  since  it  is  not  equal  to  gi'  can- 
not be  located  by  a  deflection  from  the  chord  gi  produced,  ac- 
cording to  Equation  22.  So  we  must  find  the  tangent  at  i  by 
setting  off  at  g  the  amount  kg  ~  8.72  feet  ;  then,  having  ob- 
tained the  tangent  at  i,  we  can  calculate  the  offset  inn  for  the 
_60-foot  chord  in  (by  Equation  23),  which  equals  3.14  feet,  and 


RAILROAD   LOCATION.  87 

this  brings  us  to  the  P.T.  of  the  curve.  In  order  to  find  the 
direction  of  the  tangent  at  n  we  may  either  set  off  at  i  the 
value  of  t  for  the  chord  in,  or  we  may  produce  the  chord  in 
to  q,  making  nq  =  in,  and  then  from  q  set  off  an  offset  qp  = 
mn  ;  np  will  then  be  the  direction  of  the  tangent. 

Theoretically,  we  ought  always  to  make  the  angle  between 
a  tangent  and  its  offset  =  90°,  and  between  a  chord  produced 
and  its  offset  =  90°  —  |  angle  subtended  by  the  chord  at  the 
centre  ;  but  in  ordinary  work  there  is  no  need  to  be  particular 
about  this. 

80.  To  locate  a  curve  by  ordinates  from  a  long  chord. 
—  Suppose,  as  in  Fig.  30,  we 
have  two  stations  a  and  b  given, 
we  then  have  the  length  of  the 
arc  adb,  and  so  we  can  find  C 
by  Equation  5. 

Now  if  d  is  the  middle  point 
on  the  curve  the  deflection-offset 
t  from  the  tangent  at  d  to  a  =  M, 
the  ordinate  at  d  ;  therefore,  by 
Equation  20, 

M=  fivers,     ......     (24) 


AT  being  the  mid-ordinate  to  the  long  chord  T.  The  length  of 
an  ordinate  from  the  chord  to  any  other  station  e  will  be  given 
by  the  equation 

.     .    .    (25) 


where  N  =  the  number  of  stations  measured  along  Hie  curve 
from  d  to  e  ;  and  the  distance  from  the  centre  of  the  long 
chord  at  which  m  must  be  set  off  is  given  by 


(26) 


which  is  the  same  as  Equation  21  for  the  value  of  X  in  Sec.  78. 
To  take  an  example  :  Suppose  a  is  at  Station  2  -f-  20  and  b  at 
Station  6  +  40,  then  d  will  fall  at  Station  4  -f  30.  Let  D  =  10°, 
then  C=  42°  ;  and  we  can  find  T  either  by  direct  measure- 
ment or  by  Equation  16  =  2R  sin  21°  =  411.2  feet.  Similarly 
by  Equation  23  we  find  M  =  38.1  feet, 


88  RAILROAD   LOCATION. 

If  we  then  wish  to  set  off  an  ordinate  to  Sta.  3.00,  we  have 
iV=1.3;  therefore  y  —  R  sin  13°  =  129.1  feet,  and  m,  by 
Equation  24,  =  38.1  —  R  vers  13°  =  23.4  feet 

It  is  usually  unnecessary  to  calculate  the  values  of  y,  except 
perhaps  when  near  the  ends  of  the  chord.  Thus,  in  the  above 
example,  had  we  assumed  y  —  100.2V  =130  feet,  it  would 
practically  have  made  no  difference  in  the  position  of  Sta.  300. 

If  we  have  the  length  of  the  chord  Y  given,  we  may  ob- 
tain G  directly  from  it  by  means  of  Equation  16 ;  or,  con- 
versely, when  we  know  C  we  can  obtain  Y. 

The  lengths  of  Long  Chords  subtending  arcs  up  to  6  stations 
are  given  in  Table  Y  ;  also  the  length  of  arcs  subtended  by 
100- foot  chords.  Thus,  if  C  =  20°  and  D  —  10° ;  Y  instead  of 
being  equal  to  200  feet,  really  equals  200. 254  feet,  which  is  the 
result  we  should  obtain  if  we  used  Equation  6  instead  of  Equa- 
tion 5  to  obtain  the  value  of  L.  The  middle  ordinate  may 
also  be  correctly  found  thus  : 


-^-, (27) 

and  any  other  ordinate 


-y\    .    .    .    (28) 


An  approximate  formula,  which  is  really  a  corruption  of  Equa- 
tion 27,  is  ' 

M  =       -  (nearly) (29) 


It  is  sufficiently  true,  however,  when  Y  is  small,  the  error 
on  a  20°  curve,  in  the  case  of  a  50-foot  chord,  only  amounting 
to  .002  foot.  By  comparing  Equation  29  with  Equation  23, 
we  see  that  the  rnid-ordinate  to  a  short  chord  may  be  con- 
sidered equal  to  one  quarter  the  tangential  off  set  at  a  distance 
along  the  tangent  equal  to  the  chord. 

A  convenient  method  of  locating  small  arcs  is  that  shown 


BAILROAD  LOCATION. 


89 


in  Fig.  31,  where,  having  found  M  by  Equation  24  or  29,  the 
mid-ordinate  for  the  subchord  ac  may  be  considered  equal  to 


FIG.  31. 


iJf,  and  the  ordinate  e  of  the  sub-subchord  dc  similarly  equal 
to  one  quarter  the  ordinate  at  d. 

81.  To  pass  a  curve  through  a  fixed  point,  the  angle 
of  intersection  feeing1  given. — Suppose  we  first  find  the 


FIG.  32. 


position  of  the  fixed  point  p  (Fig.  32)  with  reference  to  Aa  in 
terms  of  the  distance  Ap  and  the  angle  aAp :  then 


pAO  =  90'  - 


+      , 


and 


sin  ApO  =  sin  pAO  sec  -  ; 
m 


therefore  in  the  triangle  ApO  we  have  pO  equal 

R  =  Ap  sin  pAO  cosec  (pAO  -f-  ApO). 
ApO  always  exceeds  90°. 


90 


RAILROAD   LOCATION. 


82.  To  run  a  tangent  from  a  curve  to  any  fixed  point. 

—Let  p  (in  Fig.  33)  be  the  fixed  point,  and  a  and  b  be  any  two 


FIG. 


points  on  the  curve, — b,  however,  being  on  the  side  remote  from 
p,  yet  as  near  to  the  probable  situation  of  the  tangent-point  d 
as  is  possible.  Then  taking  the  chord  ab  as  a  base,  the  length 
of  which  is  given  by  Equation  16,  observe  the  angles  at  a  and 
b  in  the  triangle  abp  ;  then 

bp  =  ab  sin  a  cosec  apb, 

when  apb  =  180°  —  (a+  6). 

Now  if  bh  is  the  tangent  at  b,  we  know  the  angle  Jibp,  and 
can  thus  find 

eb  —  2R  sin  hbp. 
But  by  Euclid 


Thus  by  measuring  off  a  distance  bf  =  bp  —  dp  and  offsetting 
to  the  curve,  we  find  the  required  tangent-point  d. 
83.  To  connect  two  curves  by  a  tangent.— First  suppose, 


FIG.  34. 

as  in  Fig.  34,  that  both  curves  are  of  the  same  direction.  On  the 
curve  of  smaller  radius  R  select  a  point  p  slightly  more  re- 
mote from  the  other  curve  than  the  tangent-point  at  a  prob- 
ably is.  On  the  curve  of  larger  radius  R  find  a  point  p' 
which  has  its  tangent  parallel  to  the  tangent  at  p.  This  may 
be  done  by  running  a  trial-line  to  some  station  s  ;  and  then,  by 
comparing  the  direction  of  the  tangents  at  p  and  s,  we  find 
how  far  along  the  curve  from  s,  p'  will  be  situated. 


RAILHOAD   LOCATION. 


91 


Now  if  pd  is  the  tangent  at  p,  and  cb  is  perpendicular  to  pp ' , 
we  have  pea  =  dpp'  —  acb,  and 

sin^=         («-*>™*!!!<       (nearly), 

#p'  -{-  (R  —  R)  sin  dpp' 

pp'  being  obtained  by  direct  measurement;  and 

aa'  =  pp'  -}-  (R1  —  R)  sin  dpp'  —  (R  —  R)  sin  acb, 

from  which  we  can  find  the  position  of  a' . 

But  suppose,  as  in  Fig.  35,  the  two  curves  are  of  opposite 
direction. 


FIG.  35. 

Then  select  p  on  the  side  of  a  towards  the  other  curve.  Then, 
as  before,  pea  =  dpp'  —  acb;  but  in  this  case 

(R  4-  R)  vers  dpp' 


sin  acb  = 


pp  -j-  (R  -j-  R)  sin  dpp 


-  (nearly), 


and 


aa'  =  pp'  -\-  (R  -\-  R)  sin  dpp'  —  (R'  -J-  R)  sin  acb. 


The  distance  ap  should  never  exceed  100  feet  when  the 
curves  are  of  the  same  direction,  or  75  feet  when  of  opposite 
direction,  and  should  always  be  taken  as  small  as  possible. 

84.  Given  a  curve  joining1  two  tangents,  to  change  the 
P.C.  so  that  the  curve  may  end  in  a  parallel  tangent. 

Let  it  be  required  to  move  the 

P.C.  at  a  (in  Fig.  36)  so  that  the    •    • J>  x" 

curve  ab,  instead  of  ending  at  b,  will 
end  in  a  parallel  tangent,  distant 
from  the  tangent  at  b  by  the  amount 


Then,  since  it  is  simply  a.  case  of 
shifting  the  curve  bodily  in  the  di- 
rection of  the  tangent  aa' ,  we  have 

aa'  =  e  cosec  1.  FIG.  36. 

Had  a'b'  been  the  given  curve,  and  it  were  required  to  shift 


EAILROAD   LOCATION". 


it  outwards  to  the  parallel  tangent  at  b,  the  same  equation  of 
course  applies. 

85.  Suppose  we  have  such  a  case  as  that  shown  in  Fig.  87, 
where  ab  is  the  given  curve,  and  it  is  required  to  shift  it  to 
parallel  tangents  at  each  end,  as  at  a'  and  b'. 


FIG.  37. 

Then  starting  from  the  tangent  at  a,  we  can,  as  above  de- 
scribed, shift  the  curve  from  the  tangent  at  b  to  the  tangent 
at  b',  and  from_the  tangent  at  a  we  can  in  the  same  way  shift 
it  on  to  the  tangent  at  a',  which  gives  us  the  required  positions 
of  a'  and  b'. 

86.  Given  a  curve  joining  two  tangents,  to  change  the 
radius  and  the  P.C.  so  that  the  new  curve  may  end  in  a 
parallel  tangent  at  a  point  opposite  to  the  original  P.T. 

In  Fig.  38  let  it  be  required  to 
change  the  radius  of  the  curve  ab 
and  also  the  position  of  a,  so  that 
the  curve,  instead  of  ending  in  b, 
will  end  in  a  parallel  tangent  at  b' 
(V  being  directly  opposite  to  b). 
Then  if  0  is  the  centre  of  the  curve 
ab  and  R  its  radius,  and  0'  the  cen- 
tre of  the  curve  a'b'  and  R'  its 
radius,  by  Equation  11, 

Ab  =  R  exsec  /, 
and          Ab'  =  R  exsec  7; 


O'i. 


FIG.  88. 


therefore 


and 


R1  -R  = 


bb' 


exsec 


aa'  = 


Had  a'b'  been  the  given  curve,  and  it  were  required  to  shift 


RAILROAD   LOCATION. 


93 


it  outwards  to  the  parallel  tangent  at  b,  the  same  equations  of 
course  apply. 

87.  Given  a  curve  joining  two  tangents,  to  find  the 
radius  of  another  curve  which,  from  the  same  P.C.,  will 
end  in  a  parallel  tangent. 

Let  it  be  required  to  change 
the  radius  of  the  curve  ab,  so 
that  it  will  end  in  a  parallel  tan- 
gent at  V.  "\  /  "V.  .-** 

Let  0  be  the  centre  of  the 
curve  ab  and  R  its  radius,  and  0' 
be  the  centre  of  the  curve  ab'  and 
R  its  radius.  Then  R  -  R  = 
00;  therefore 


R  -  R  =  - 


vers  /' 


FIG.  39. 


Had  ab'  been  the  given  curve,  and  it  were  required  to  shift 
it  outwards  to  the  parallel  tangent  at  b,  the  same  equation  of 
course  applies. 

88.  Given  a  curve  joining  two  tangents,  to  change  the 
radius  and  position  of,  the  P.C.  so  that  the  curve  may 
end  in  the  same  P.T.,  but  with  a  given  change  in  direc- 
tion. 

In  Fig.  40  let  it  be  required  to 
change  the  radius  and  P.C.  of  the 
curve  ab,  so  that  at  b  it  will  have  a 
difference  in  direction  equal  to  /'  —  I. 
Then  if  0  is  the  centre  of  the  curve 
ab  and  R  its  radius,  and  0'  and  R'  are 
the  centre  and  radius  of  the  curve  a'b, 

Myers  I  =  R'  vers/'; 
therefore 


D,      R  vers  7 

=  - 


FIG-40- 


and  aa'  =  R  sin  /—  R'  sin  /', 


94 


RAILROAD   LOCATION. 


COMPOUND  CUKVES. 

89.  A  compound  curve,  being  merely  a  series  of  two  or  more 
simple  curves,  the  manner  in  which  it  is  located  is  by  setting 
out  its  components  separately,  each  P.C.C.  (Point  of  Com- 
pound Curvature)  being  treated  as  a  P. C.  or  P.T.,  the  direction 
of  the.tangent  at  each  P.C.C.  being  given  by  its  Index-reading. 

As  regards  the  notes,  instead  of  keeping  them  for  each 
curve  independently,  it  is  better  to  carry  the  Index-reading 
through  continuously  from  the  P.C.  to  the  P.T.,  so  that  the 
reading  for  the  P.T.  equals  half  the  total  intersection-angle. 

The  length  and  intersection-angle  of  eacJi  component  curve 
should  be  entered  in  the  notes,  and  also  the  total  length  and 
total  intersection-angle. 

90.  To  locate  a  compound  curve  when  the  P.C.C.  is 
inaccessible. 


FIG.  41. 

Suppose,  as  in  Fig.  41,  p  (the  P.C.C.)  is  inaccessible.  The 
points  e  and  d,  if  accessible,  may  then  be  found  by  inserting 
the  value  of  the  intersection-angle,  in  the  case  of  each  curve 
separately,  in  Equation  9,  and  thus  obtaining  for  T  the  dis- 
tances ad  and  be. 

Then  from  the  tangent  de  the  curve  can  be  located  by  offsets, 
as  already  shown. 

If  the  points  d  and  e  are  also  inaccessible,  select  in  the  curve 
some  convenient  point  /,  and  from  it  set  oft'  the  offset/ft  = 


RAILROAD   LOCATION". 


95 


of  vers  fop  (by  Equation  20).  Similarly,  from  a  point  in  the 
other  branch  of  the  curve  lay  off  an  offset  ik  —  qi  vers  iqp. 
We  can  then  rind  the  position  of  p  by  Equation  21 ;  thus : 

hp  =  of  sin  fop. 

91.  Given  a  simple  curve  ending  in  a  tangent,  to  con- 
nect it  with  a  parallel  tangent  by  means  of  another 
curve. 

1.  Let  ac  in  Fig.  42  be  the 
given  curve,  and  be  the  required 
curve:  then  we  have 


cos  C  =  1  — 


R-r' 


p.c.c. 


FIG.  42. 


from  which  we  can  at    once 
find  the  P.C.C. 

2.  Let  be  be  the  given  curve, 
and  ac  the  required  curve:  then  since  C,  the  central  angle,  is 
the  same  for  both  curves,  the  above  equation  holds  good  also 
in  this  case. 

92.  To  connect  a  curve  with  a  tangent  by  means  of 
another  curve  of  given  radius, 


-JL 


p.c.c 


FIG.  43. 


1.  Let  ac  in  Fig.  43  be  the  given  curve  which  it  is  required 
to  connect  with  a  given  tangent  at  b.  Find  the  point  a  on  the 
given  curve  which  has  its  tangent  parallel  to  the  given  tan- 
gent, and  measured:  then,  since 


cos  C  =  1  — 


R-  r' 
we  can  thus  find  the  position  of  the  P.C.C. 


96 


EAILROAD   LOCATION. 


2.  But  if  the  radius  of  the  required  curve  is  less  than  that  of 
the  other  curve,  then,  as  in  Fig.  44,  find  the  point  dj&t  the 
intersection  of  the  tangent  at  b 
with  the  given  curve  ac,  and  ob- 
serve the  angle  of  intersection  at 
d  =  aod;  then 

R  cos  (aod)  —  r 


cos  aop  = 


R-r 


Thusj9,  the  P.C.C.,  will  be  sit- 
uated at  a  distance  along    the 
FlG-  44-  curve  from  d  represented  by  the 

curvature  aop  —  aod. 

3.  An  analogous  case  is  that  shown  in  Fig.  45,  where^it  is 
required  to  connect  the  curve  ac  with  a  tangent  on  the  convex 
side  by  means  of  the  curve  pb. 

Then,  as  before,  find  d  and  observe  the  angle  of  intersection 
at  d  =  aod ;  then 


cos  (aop)  = 


R  cos  (aod)  —  r 


from  which  we  can  find  p  as  above. 


FIG.  45. 

Suppose  in  case  3  the  point  d  were  found  to  coincide  with 
a;  then  we  merely  have  the  case  of  a  Y  located  on  the  tangent 
db,  in  which  case  the  above  formula  becomes 


cos  (aop)  — 


R-r 


RAILROAD   LOCATION. 


97 


93.  Given  a  compound  curve  ending  in  a  tangent,  to 
change  the  P.C.C.  so  that 
the  terminal  curve  may  end 
in  a  given  parallel  tangent 
without  changing  its  ra- 
dius. 

1.  In  Fig.  46  let  the  radius 
of  the  terminal  curve  pb  be 
greater  than  the  radius  of  the 
other  curve-pa  ;  then, 

A..  If  we  want  to  shift  tJie 
curve  inwards  to  b' ,  then  to  find 
p1,  the  new  position  of  the 
P.C.C.,  \ve  have 


FIG.  46. 


cos  o  =  cos  o  -\- 


R-r' 


but, 

B.  If  apb'  were  the  given  curve,  and  it  were  required  to  shift 
it  outwards  to  b,  then 


cos  o  =  cos  o  — 


R-r' 


and  since  in  both  cases 


pqp'  =0-0', 


we  can  thus  find  the  position  of  p  or  p',  as  the  case  may  be. 

2.  Suppose,  however,  the  radius 
of  the  terminal  curve  bp  is  less 
than  the  radius  of  the  other  curve 
pa  as  in  Fig.  46,  and  that  it  is 
required  to  shift  the  tangent  (A) 
inwards  to  b:  then 


cos  o  =  cos  o  —  - 


R-r 


But   (B)  if   ap'b'  were  the  given  FIG.  47. 

compound  curve,  and  it  were  required  to  shift  it  outwards, 
then 


cos  o  =  cos  o'  -f-  -, 


R-r 


98 


RAILROAD   LOCATION. 


Then  since  in  both  cases  (A)  and  (B)  pqp'  =  o'  —  o,  we  can 
find  the  position  of  p  or  p'  as  the  case  may  be. 

94.  To  connect  two  curves,  already  located,  by  means 
of  another  curve  of  given  radius. 


As  in  Fig.  48,  let  E  be  the  radius  of  the  easier  curve,  and  r 
the  radius  of  the  sharper  curve.  Find  the  tangent  ab  as 
shown  in  Sec.  83,  and  also  the  distance  ab  by  direct  measure- 
ment or  calculation ;  then 


and 


tan  (aqs)  = 


qs  =  ab  cosec  (aqs). 


Then,  since  oq  =  op  —  R  and  os  =  op'  —  r,  where  op  and  op' 
are  each  equal  to  the  radius  of  the  required  curve,  we  have 
the  three  sides  of  the  triangle  oqs,  from  which  we  can  find 
the  angle  oqs  (see  Sec.  231);  and 

aqp  —  180°  —  (oqs  +  aqs). 

Thus  we  can  find  the  position  of  p. 

Similarly,  we  can  find  the  position  of  p'-,  or  we  can  calculate 
the  angle  at  o,  which  does  equally  well. 

The  radius  of  the  required  curve  must  exceed 


If  R  =  r,  then 


RAILROAD   LOCATION. 


99 


95.  To  locate  any  portion  of  a  compound  curye  from 
any  station  on  the  curve. 


FIG.  49. 

Let  abce  in  Fig.  49  be  a  compound  curve,  and  a  any  station 
on  the  curve,  and.let  it  be  required  to  establish  the  point  e ; 
the  P.C.C.'s  at  b  and  c  being  inaccessible. 

Assume,  -for  the  sake  of  simplicity,  that  the  chords  ab,  be, 
and  ce  are  equal,  and  let  the  curvature  of  be  equal  twice  the 
curvature  of  ab,  and  that  of  ce  three  times  the  curvature  of 
ab. 

Now  if  d  —  the  deflection  from  the  tangent  at  a  for  Sta.  b, 
then,  if  ab  be  produced  to  /,  the  angle  fbc  =  d  -\-  2d  =  3d. 
Again,  if  the  chord  be  be  produced  to  g,  the  angle  ecg  =  2d  -f  3d 
=  5d.  Then  in  the  triangle  abe,  the  angle  at  b  —  180°  —  M ; 
and  since  the  length  of  the  chords  can  be  found  by 
Equation  16  (Sec.  74),  we  can  find  the  side  ae  and  the  angles 
at  a  and  c.  Again,  in  the  triangle  ace,  the  angle  at  c  —  180°  — 
(bca  -f-  5d);  thus  we  can  find  the  angle  at  a.  Similarly  we  can 
find  the  angle  subtended  at  a  by  the  chord  be,  and  thus  we 
have  the  total  deflections  to  b,  c,  and  e.  When  the  chords  are 
of  different  lengths,  as  is  of  course  usually  the  case  in  prac- 
tice, and  the  curvature  varies  irregularly,  we  can  by  plotting 
the  curves  and  drawing  the  tangent  at  each  P.C.C.  see  at  once 
in  each  case  what  the  deflection-angle  at  any  P.C.C.  will  be 
from  the  chord  produced.  The  principle  will  be  just  the 
same  as  in  the  case  above  described. 

Sec.  96  is  an  application  of  this  problem. 


100  RAILROAD   LOCATION. 


TRANSITION   CURVES. 

96.  Since  the  elevation  and  depression  of  the  outer  and 
inner  rails,  respectively,  at  the  entrance  to  a  curve  must  be 
made  gradually,  and  for  any  given  speed  the  difference  in 
elevation  varies  inversely  as  the  radius  of  curvature,  it  fol- 
lows that  the  curvature  should  also  decrease  gradually,  having 
a  radius  equal  to  infinity  at  the  P.O.  and  a  minimum  at  the 
centre  of  the  curve.     If  we  assume,  as  is  usual,  that  the  dif- 
ference in  elevation  of  the  two  rails  increase  at  a  uniform  rate 
until  the  maximum  curvature  is  attained,  then  the  theoretic 
curve  which  should  be  adopted  is  a  form  of  the  elastic  curve, 
which,  on  account  of  the  trouble  involved  in  locating  it,  has 
been  supplanted  by  various  approximations,  such  as  the  curve 
of  sines,  parabolae,  etc.;  these  being  easier  to  locate  in  the 
field. 

The  use  of  Transition  Curves  is  found  not  only  to  cause 
less  resistance  to  the  passage  of  trains  than  a  similar  curve 
whose  ends  are  not  eased  off,  but  also  generally  to  enable  the 
curves  to  be  fitted  better  to  the  ground  than  in  the  case  of  plain 
circular  ones. 

That  Transition  Curves  are  of  advantage  in  actual  practice 
is  shown  by  the  fact  that  all  Simple  Curves  at  their  P.C.'s  and 
P.T. 's  have  a  decided  tendency  to  assume  the  form  of  the 
Elastic  Curve  ;  and  since  this  lateral  creeping  is  caused  by  the 
pressure  of  the  flanges  of  the  wheels,  increased  wear  and  tear 
to  rails  and  rolling-stock  is  the  result. 

It  is  to  be  noticed  that  the  easing  of  curves  in  many  cases 
involves  an  increase  in  curvature  at  the  centre  of  the  curve,  but 
this  is  usually  so  slight  as  to  be  practically  inappreciable,  and 
is  much  more  than  compensated  for  by  the  reduction  of  curva- 
ture at  the  ends  of  the  curve.  Thus,  for  example,  where  a  9° 
simple  curve  defines  the  limit  of  curvature  in  the  case  of 
uneased  curves  on  any  road,  by  inserting  transition  curves  a 
10°  curve  would  be  perfectly  allowable. 

The  three  following  methods  of  inserting  transition  curves 
are  simple  and  easily  applied: 

97.  Method  I. — Suppose,  as  in  Fig.  50,  that  we  have  a 
5°  30'  curve  db,  which  it  is  required  to  ease  off  by  means  of  a 
transition  curve. 


RAILROAD   LOCATION. 


101 


Now  if  we  do  not  wish  to  shift  the  main  curve  inwards  from 
the  tangent  at  a,  it  becomes  necessary  to  shift  the  tangent  at  a 


0.50 1 "^-^^ 


FIG.  50. 

itself  outwards  by  the  amount  ac,  and  also  to  throw  the  P.O. 
at  a  backwards  by  the  amount  oc,  so  that  the  point  o  becomes 
the  new  P.O. 
Now 

ac  =  T  sin  d  —  R  vers  0, 
and 

oc  =  Tcos  d  —  R  sin  C,     . 

where  Y—  the  long  chord  to  the  end  of  the  transition  curve; 
d  =  the  total  deflection-angle  from  Sta.  o  to  the  end  of  the  tran- 
sition curve  (given  in  top  line  of  Tables  A  and  B  in  this  sec- 
tion); C  =the  total  curvature  of  the  transition  curve,  as  rep- 
resented by  the  angle  esa  (values  of  which  are  given  in  Tables 
A  and  B);  and  R  =  Radius  of  the  main  curve. 

The  values  of  the  first  term  in  each  of  these  equations  are  also 
given  (i.e.,  T"sin  d  and  Fcos  d)  in  Tables  A  and  B. 

Suppose  we  consider  that  a  transition  curve  which  increases 
its  curvature  by  1°  in  every  50  feet  (as  in  Table  A)  will  suit  the 
case  in  question,  then  we  want  250  feet  of  such  a  curve  in  order 
that  the  increase  in  curvature  at  no  point  may  exceed  1°,  and  in 
that  case  we  find  from  the  above  formula  that  oc  =  113.40  feet 
and  ac  =  3.06  feet;  so  that  the  tangent  must  be  offsetted  to  the 


102 


KAILROAD   LOCATION. 


left  a  distance  of  3.06  feet,  and  the  new  P.O.  will  be  situated 
113.40  feet  back  from  the  original  one. 

Set  the  transit  up  at  the  point  o  and  locate  the  curve  in  the 
usual  manner,  the  zero  of  the  instrument  coinciding  with  the 
direction  of  the  tangent,  the  index  readings  being  taken  from 
the  top  line  in  Table  A.  The  point  e  at  Sta.  2.50  from  o  will 
then  be  the  P.C.C.  of  the  5°  branch  of  the  transition  curve  and 
the  5°  30'  main  curve.  Should  the  point  e  not  be  visible  from 
o,  the  transit  may  be  moved  up  to  any  of  the  intermediate  sta- 
tions, and  the  total  deflection  for  the  other  stations  from  the 
tangent  at  any  station  are  given  in  the  tables;  so  that,  suppose 
we  had  found  it  necessary  to  move  up  to  Sta.  1.50,  then  we  can 
get  the  zero  of  the  instrument  to  coincide  with  the  direction  of 
the  tangent  at  that  station,  by  setting  the  vernier  to  the  deflec- 
tion for  Sta.  1.50  (taken  from  the  top  line  in  the  table)  when 
the  telescope  is  clamped  on  to  the  back-sight  at  Sta.  o.  We 
then  proceed  as  before;  e.g.,  our  index-reading  for  e  will  be 
3°  25',  and  so  on. 

Had  a  change  of  1°  in  every  50  feet  extended  the  transition 
curve  too  much,  we  might  have  adopted  the  curve  given  in 
Table  B. 


TABLE  A.— CHANGING  1°  IN  EVERY  50  FEET. 

TOTAL  DEFLECTIONS  PROM  THE  TANGENT  AT  ANY  STATION, 
AND  THE  VALUES  OF  C,  T  sin  d,  AND  Y  cos  d. 


0 

.50 

1.00 

1.50 

2.00 

2.50 

3.00 

Transit. 
0°  15' 
0°  52*' 
\°W 
3°  07*' 
4°  45' 
6°  42*' 

0°  15' 
Transit. 
0°  30' 

1°  22*' 
2°  35' 
4°  07*' 
6°  00' 

0°  37*' 
0°30' 
Transit. 
0°45' 
1°  52*' 
3°  20' 
5°  07*' 

1°  10' 

1°07* 
0°  45' 
Transit. 
1°00' 
2°  22*' 
4°  05*' 

1°  52*' 
1°55' 
1°  37*' 
1°00' 
Transit. 
1°  15' 
2°  52*' 

2°  45' 
2°  52* 
2°  40' 

?gr 

Transit. 
1°  30' 

3°  47*' 
4°  00' 
3°  52*' 
3°  25' 
2°  37*' 
1°30' 
Transit. 

C 

0°30' 

1°  30' 

3°  00' 

5°  00' 

7°  30' 

10°  30' 

Fsind 
in  feet. 

0.22 

1.09 

3.05 

6.54 

11.98 

19.80 

Ycosd 
in  feet. 

50.00 

99.99 

149.95 

199.81 

249.41 

298.74 

RAILROAD  LOCATION.  103 

TABLE  B.— CHANGING  2°  IN  EVERY  50  FEET. 


0 

.50 

1.00 

1.50 

2.00 

2.50 

3.00 

Transit. 
0°30' 
1°45' 
3°  40' 
6°  15' 
9°  30' 
13°  25' 

0°30' 
Transit. 
1°00' 
2°  45' 
5°  10' 
8°  15' 
12°  00' 

1    15' 

1    00' 
Transit. 
1    30' 
3    45' 
6    40' 
10°  15' 

2°  20' 
2°  15' 
1°30' 
Transit. 
2°  00' 
4°  45' 
8°  10' 

3°  45 
3°  50 
3°  15' 
2°  00' 
Transit. 
2°  30'     ' 
5°  45' 

5°  30' 
5°  45' 
5°  20' 
4°  15' 
2°  30' 
Transit. 
3°00/ 

7°  35' 
8°  00' 
7°  45' 
6°  50' 
5°  15' 
3°  00' 
Transit. 

C 

1°00' 

3°  00' 

6°  00' 

10°  00' 

15°  00' 

21°  00' 

Fsind 
in  feet. 

0.44 

2.18 

6.10 

13.06 

23.89 

39.37 

Fcos  d 
in  feet. 

50.00 

99.98 

149.80 

199.32 

248.12 

295.70 

The  stations  located  as  above  need  only  be  considered  as 
ternpwary  ones,  by  means  of  which  the  true  stations  may  be 
located.  These  may  be  best  obtained  as  follows:  Suppose 
Sta.  o  falls  really  at  Sta.  304  +  34,  then  Sta,  304  +  50  can  be 
located  by  stretching  a  tape  between  temporary  Stations  o  and 
0.50  and  setting  oft'  the  ordinate  if  (Equation  24,  Sec.  80)  16  feet 
along  it  from  o,  and  so  on  between  the  different  stations. 
Values  of  M  are  given  in  the  following  table  for  a  1°  curve. 
The  value  of  M  for  any  other  curve  may  be  considered  to  vary 
as  the  curvature,  so  that,  for  example,  for  a  9°  curve  the  ordi- 
nate at  any  point  will  be  9  times  that  given  in  the  table  for  the 
corresponding  distance. 

VALUES  OF  M  FOR  1°  CURVE,  50-FT.  CHORDS. 


Dist.  from 
Temp.  Sta. 

M  in  feet. 

Dist.  fiom 
Temp.  Sta. 

M  in  feet. 

Dist.  from 
Temp.  Sta. 

M  in  feet. 

2  ft. 

4  " 
6  " 
8  " 

.011 
.016 
.022 
.030 

10ft. 
12  " 
14  " 
16  " 

.035 
.040 
.044 
.048 

18ft. 
20  " 
22  " 

24  " 

.050 
.052 
.054 
.054 

The  principal  objection  which  can  be  urged  against  this 
curve  is  its  rigidity;  this  is  in  a  great  measure  overcome  by 
having  the  option  of  the  two  sets  of  curves  given  above,  one 
changing  by  1°  every  50  feet,  and  the  other  by  2°.  Generally 
speaking,  the  former  is  adapted  to  curves  not  exceeding  7°,  and 


104 


BAILROAD  LOCATION". 


the  latter  to  curves  of  from  6*  to  14°  curvature;  while  for  curves 
of  from  5°  to  8°  either  set  may  be  employed. 

Another  objection  which  may  be  brought  against  it,  and  one 
which  is  often  brought  against  transition  curves  generally,  is 
that  it  is  not  worth  the  trouble  taken  in  locating  it.  As  regards 
this,  the  use  of  transition  curves,  not  only  theoretically  but 
practically,  is  found  to  reduce  the  resistance  of  the  curve  very 
materially,  to  lessen  the  cost  of  maintenance  of  way,  to  reduce 
the  chances  of  derailment,  and  considerably  to  ease  the  motion 
of  the  cars. 

There  is  no  need  to  set  out  the  transition  curves  during  the 
location,  but  the  tangent  in  any  instance  should  be  run  to  c 
(Fig.  50)  and  the  transit  then  offsetted  to  a,  from  which  point 
the  main  curve  can  be  located.  The  amount  of  the  offset  ac, 
and  the  distance  oc,  should  be  added  to  the  notes  of  the  curve, 
and  also  the  distance  ae,  which  represents  G.  The  general  plan 
of  the  location  then  shows  the  curves  as  in  Fig.  16.  Then  when 
the  engineer  takes  charge  of  the  work  for  construction  he  has 
simply  to  "reference"  the  points  o  and  e,  and  run  in  the  curve 
by  means  of  the  above  table,  as  easily  as  he  would  any  simple 
curve. 

98.  Method  II. — Another  form  of  transition  curve  is  that 
shown  in  Fig.  51.  It  is  especially 
suitable  in  cases  where  it  is  more 
convenient  to  offset  the  curve  than 
the  tangent  itself.  It  practically 
converts  the  original  simple  curve 
into  a  3-centre  one,  but  where  the 
curvature  of  the  main  curve  is  light, 
it  answers  the  purpose  of  easing 
off  the  curvature  at  its  ends  suf- 
ficiently in  ordinary  cases. 

In  Fig.  51,  let  r  =  radius  of  the 
FIG.  51.  original  main  curve  ab. 

Offset  ab  inwards  by  an  amount  af  =  e;  then  if  R  —  radius 
of  the  terminal  curve  cd,  we  have 


cosfod  =  1  — 


from  which  we  can  find  the  position  of  d  ;  and 

ca  =  R  — •  (r  —  e)  siufod, 


RAILROAD  LOCATIOH.  105 

from  which  we  can  find  the  position  of  c.    The  curve  cd  can 
then  be  best  located  with  a  transit  from  the  point  c. 

A  convenient  method  of  applying  this  principle  in  practice 
is  to  make  e  =  0.2  foot  for  every  degree  of  curvature  of  ab,  and 
to  make  R  —  3(r  —  e);  then  if  we  make/<#  =  33.9  feet,  d  is  the 
P.C.C.,  and 

ca  =  2(r  —  e)  sin/otf, 

fod  being  found  from  the  formula 

e 


cos  fod  =  1  — 


2(r  -  e) 


For  ordinary  curves  ca  then  varies  from  75  to  100  feet. 

99.  Method  III. — Another  method  of  substituting  a  3-centre 
curve  for  a  simple  one,  when  we  do  not  wish  to  change  the 
original  tangent-points,  is  as  follows: 

In  Fig.  52  let  o  be  the  centre  of  the  original  simple  curve  afb, 
the  radius  of  which  =  R  ;  and  let  Oi  be  the  centre  of  the  new 
main  curve  ced,  whose  radius  =  Rl .  And  let  02 ,  #2  be  the 
centre  of  the  terminal  curves  ac  and  db,  whose  radii  =  J?2 . 


°2 

FIG.  52. 

1.  Given  Ri  and  R* . 
Then 


(*,-*)  sin 


and 

aob  — 


106  BAILROAD  LOCATIOK. 

Thus  we  obtain  the  position  of  the  points  c  and  d. 

2.  Given  Ri  and  ao-tC  =  bo^d. 

Then 

adb  coid 

R  sin  —  --  Rl  sm  -~ 

R,  =  --  *  --  _?__. 

.    aob         , 

sin  -  --  sin 


The  curvature  of  the  arc  ced  should  never  exceed  that  of  ab 
by  more  than  1°  (about  50'  excess  is  usually  a  suitable  amount), 
and  Ri  should  equal  about  3R. 

The  distance 

fe  =  (R^-Rl)  sin  ao*c  cosec  ^~  -  (R  -  BJ. 


Suppose,  however,  in  substituting  the  3-centre  curve  for  the 
simple  one,  it  is  advisable  for  the  points  e  and/ to  coincide  as 
in  Fig.  53. 


1.  Given  Hi  and  Rz ,  we  then  have 

(R -Rivers  ~ 
vers  uo&  =  —  — » 5 


Then  a  must  be  put  back  on  the  tangent  to  u,  and 

aob  I       uorf  aob\ 

<w  =  (R  —  Ri)  vers  —  I  cot  -~-  —  cot  — I. 


RAILROAD  LOCATION. 
2.  Given  Hi  and  uo^c,  we  then  have 


107 


au  being  found  as  above. 


VERTICAL  CURVES. 

100.  We  have  already  considered  the  dangers  which  arise 
from  sudden  changes  of  grade  (see  Sec.  29).  Where  these 
changes  are  considerable,  amounting  to,  say,  0.5  p.  c.  in  the 
difference  of  grade,  it  is  advisable  to  round  off  the  angle  at 
the  junction  of  the  two  grades  by  means  of  vertical  curves. 
On  bridge-work  this  should  be  more  especially  attended  to. 
Theoretically,  the  curve  which  should  be  applied  is  a  parabola, 
and  this  happens  also  to  be  the  simplest  form  of  curve  to 
insert  in  practice. 


In  Fig.  54  let  ac  and  cb  be  two  grades  between  which  it  is 
required  to  insert  a  vertical  curve. 

Now  cf=2cd;  therefore,  if  the  letters  a,  b,  and  c  stand 
respectively  for  the  elevations  at  those  points, 


and  the  correction  e  at  any  other  point  is  given  by  the  equa- 
tion 


_ 

c 


ac  and  cb  are  usually  made  about  200  feet  each. 


108  RAILROAD   LOCATION. 

Vertical  curves  are  not  usually  inserted  during  location,  or 
even  shown  on  the  location  profile;  but  the  corrections  for 
them  should  be  worked  out  before  the  cross-sectioning  begins, 
and  the  grade  as  shown  on  the  construction  profile  should  be 
the  corrected  grade. 


Note.— In  dealing  with  deflection-angles  and  offsets  of  curves,  the  en- 
gineer— entirely  ignorant  of  the  Differential  Calculus— may  often  save 
himself  a  considerable  amount  of  labor  by  making  use  of  the  principle 
of  Successive  Differences,  an  application  of  which  is  given  in  Sec.  203, 
Part  III.  Thus,  e.g.,  the  deflection-angles  given  in  Tables  A  and  B,  Sec. 
97,  may  be  calculated  up  to  300  feet  merely  by  the  application  of  the 
2d  differences,  and  may  be  extended  considerably  beyond  that  amount  by 
using  the  3d  differences.  More  especially  is  this  method  applicable  in 
calculating  offsets  to  a  curve  which  may  be  considered  to  vary  as  the 
Square  of  the  tangential  distance,  for  then  their  2d  differences  will  be 

constant.    As  an  example  of  this,  the  values  of  (H  —  H')a  r^ — g,  given 

in  Sec.  130,— varying  as  the  square  of  (H  —  H'),— have  for  their  2d  differ- 
ence 1.852,  which  does  not  change;  therefore  the  differences  of  the 
differences  of  the  values  in  the  table  increase  regularly,  the  difference 
between  any  two  values  being  greater  than  the  preceding  difference  by 
this  amount;  thus  the  calculation  of  such  a  table  as  that  is  merety  a 
matter  of  simple  addition  as  soon  as  the  2d  difference  has  been  obtained. 
The  engineer  should  be  always  on  the  lookout  for  this  in  the  construc- 
tion of  tables,  etc. 


PART  II. 
CONSTRUCTION. 


101.  THE  Field-work  of  engineering  during  Construction 
may  be  divided  into  two  parts,  the  first  (A)  dealing  with  the 
setting  out  of  the  work,  and  the  second  (B)  with  the  estimat- 
ing of  the  labor  and  material  employed  in  its  execution;  and 
in  this  order  it  will  be  well  to  consider  the  subject. 

A.  THE  SETTING  OUT  OF  WORK. 

102.  An  engineer,  when  given  a  subdivision  of  a  road  to 
look  after  during  its  construction,  often  finds  merely  the  centre- 
line staked   out  at  every  100  feet, — with  hubs  indicated  by 
Guard-stakes  at  the  transit  stations, — and  bench-marks  every 
half-mile  or  so  apart.     He  is  provided  with  a  copy  of  the  loca- 
tion profile  and  of  the  transit-notes  and  bench-marks,  and  with 
the  notes  and  plans  connected  with  any.special  features  in  the 
construction  on   his  subdivision   for  which  he  will   be  held 
responsible— such  as  plans  of  bridge-sites,  culverts,  etc. 

If  in  a  timber  country,  the  first  thing  he  has  to  do  is  to  see 
to  the  Clearing  of  the  Right  of  Way,  which  he  does  by 
marking  out  the  limits— if  the  clearing  is  to  be  carried  to  the 
full  width— by  blazing  the  trees  at  distances  of  a  hundred  feet 
or  so  apart  on  either  side  of  the  centre-line,  and  inscribing  the 
letter  C. 

While  the  clearing  is  being  done,  he  usually  has  time  to 
examine  the  country  along  the  line  with  an  eye  to  the  location 
of  culverts  and  the  size  of  openings  necessary,  and  to  make  a 
closer  examination  of  the  probable  classification  of  the  cuts 
than  the  location  party  probably  had  the  opportunity  of  doing. 

103.  In  order  to  obtain  a  correct  idea  as  to  what  size  of 
openings  may  be  necessary,  he  is  guided  by  the  flood-marks 

109 


110  KAILKOAD    CONSTRUCTION. 

along  the  water- courses;  and  if  there  is  any  doubt  about  these 
in  the  neighborhood  of  the  line,  he  must  follow  them  up  until 
he  finds  some  definite  indication  of  the  amount  of  flow,  or  else 
forms  a  more  or  less  accurate  estimate  of  it  for  himself,  by  an 
examination  of  its  source. 

In  selecting  the  points  for  culverts  and  the  sizes  required, 
the  engineer  must  bear  in  mind  the  effect  of  drainage  upon 
the  natural  well-defined  water-courses:  for  instance,  water  that 
before  the  construction  of  ditches  ran  more  or  less  broadcast 
over  the  country,— as  is  frequently  the  case  in  low  marshy 
land, — thereby  perhaps  in  a  dry  season  showing  no  indications 
of  its  existence  at  another  time  of  the  year,  or  which  in  a  wet 
season  may  be  simply  indicated  by  a  saturation  of  the  soil, 
may,  when  conducted  by  ditches  to  the  mouth  of  a  culvert, 
present  a  very  decided  reality. 

Often  too,  by  cutting  a  small  ditch,  two  streams  can  be 
brought  together  at  a  less  cost  than  would  be  involved  by  the 
construction  of  two  separate  culverts.  For  a  masonry  culvert 
is  an  expensive  article  in  the  first  place,  and  the  usual  substitute 
— a  timber  one — a  still  more  expensive  article  in  the  long  run. 
When  the  dump  is  low,  open  wooden  culverts  are  the  best  to 
use  as  temporary  expedients,  for  any  defects  in  them  are 
readily  visible,  and  masonry  culverts  can  be  built  to  replace 
them  with  very  little  trouble.  For  small  openings  piping  does 
admirably,  but  should  be  well  bedded;  as  a  temporary  sub- 
stitute for  pipes,  small  plank  culverts  may  be  inserted,  which 
may  afterwards  serve  as  a  means  of  inserting  the  pipes  them- 
selves. 

104.  A  thorough  system  of  drainage  along  each  side  of  the 
road-bed  should  be  one  of  the  first  points  to  which  the  attention 
of  the  engineer  should  be  given,  for  it  is  often  possible  to 
greatly  decrease  the  cost  of  construction  by  constructing 
ditches  some  little  time  before  the  commencement  of  the  work. 

As  regards  the  form  and  size  of  such  ditches,  it  is  usually 
sufficient  to  make  them  with  slopes  of  1  to  1,  but  with  plenty 
of  width  in  the  base:  as  a  rule,  for  each  foot  of  water  likely  to 
be  in  the  ditch  there  should  not  be  less  than  three  feet  of  base, 
and  the  rate  of  fall  should  be  made  as  uniform  as  is  compatible 
with  the  cost  of  construction.  For  small  ditches,  the  rate  of 
fall  should  not  be  less  than  0.2  p.  c.  if  possible;  but  a  large 
ditch  which  is  likely  to  have  a  depth  of  water  of  not  less  than 


EAILROAD   CONSTRUCTION.  Ill 

one  foot  wih  draw  tolerably  well  with  a  fall  of  only  0.1  p.  c. 
Neither  should  the  fall  be  so  great  as  to  permit  scouring  to 
any  large  extent. 

Small  extra  ditches  are  usually  staked  out  with  centre-stakes 
only,  and  the  amount  of  excavation  calculated  from  the  centre- 
heights.  But  for  larger  ones  slope-stakes  should  be  set,  and 
if  the  surface  is  irregular  it  must  be  properly  cross- sectioned. 

105.  It  is  often  the  case  that  the  cross-sectioning  of   the 
work  has  been  done  by  a  party  detached  from  the  main  location 
party:  if  so,  the  engineer  usually  has  time  to  check  the  bench- 
marks and  insert  new  ones  for  himself  at  points  which  he  may 
consider  suitable.     These  B.M.'s  should  not  be  less  than  10 
stations  apart;  their  positions  should  be  such  as  to  do  away  as 
mnch  as  possible  with  turning-points.     They  should  be  marked 
B.M.,  and  the  elevation  of  each  inscribed  on  it.     At  each 
bridge-site  there  should  be  a  bench-mark  close  at  hand.     It  is 
a  good  plan  also,  if  there  is  time,  to  check  the  alignment  from 
the  transit-notes.     Any  error  discovered,  either  in  the  levels 
or  the  alignment,  should  be  at  once  reported.     For  discrepan- 
cies arising  in  the  checking  of  the  alignment  by  using  short 
chords,  see  Part  I. 

106.  When,   however,  the  subdivision   engineer  has  the 
cross-sectioning  to   do   himself,  if  the   construction   is  being 
started  at  various  points  on  his  work  almost  simultaneously 
with  his  taking  charge,  he  then  has  his  time  from  the  very  first 
fully  occupied  in  taking  cross  sections. 

The  amount  of  work  which  this  involves  depends  a  good 
deal  on  the  manner  in  which  the  grading  is  to  be  measured. 
If  measured  in  excavation  only,  then  it  is  merely  the  cuts  that 
have  usually  to  be  cross-sectioned;  but  if  measured  in  cut  and 
fill,  both  must  receive  equal  attention.  In  the  former  case, 
where  borrowing  has  to  be  done,  it  is  often  necessary,  however, 
to  have  the  fills  also  cross-sectioned,  for,  owing  to  the  impossi- 
bility of  measuring  the  borrow-pits  correctly,  the  work  may 
have  to  be  measured  in  the  fills,  and  this  must  be  borne  in 
mind  at  the  time  of  cross-sectioning.  Also,  to  obtain  a  collect 
estimate  of  the  over-haul  it  is  necessary  to  have  the  fill 
connected  with  it  cross-sectioned.  At  all  points,  too,  where  the 
question  of  the  distribution  of  material  is  likely  to  arise,  cross- 
sections  of  the  fills  are  useful,  but  these  need  not  betaken  with 


KAILHOAD    COXSTRUCTIO^. 

the  same  accuracy  as  those  required  for  the  measurement  of 
the  work. 

To  cross-section  properly,  five  men  are  wanted  besides  the 
engineer, — namely,  a  rodman,  a  man  to  carry  stakes,  another 
to  drive  them  and  another  to  mark  them, and  a  tapeman, — for 
though  the  setting  of  slope-stakes  is  sometimes  done  separately 
from  the  cross- sectioning,  it  usually  saves  both  time  and  expense 
to  do  both  at  once. 

Before  starting  to  cross-section,  the  engineer  will  do  well  to 
construct  a  small  table  for  each  different  width  of  road-bed  aud 
set  of  slopes  which  he  is  likely  to  use,  giving  the  "  distances 
out  "  to  the  slope-stakes  for  various  amounts  of  side-heights. 
For  though  he  rapidly  acquire  these  after  a  little  practice, — 
and  should  be  checked  in  his  calculations  of  them  by  the 
rodman,— still,  by  having  a  table  before  him,  he  saves  con- 
siderable mental  work  and  insures  greater  accuracy.  He 
should  also  be  provided  with  a  small  scratch-block. 

The  best  way  to  explain  the  method  of  cross-sectioning  is  by 
means  of  an  example. 


Let  bBAG,  in  Fig.  55,  represent  a  surface  which  we  wish  to 
cross-section.  We  first  take  the  elevation  at  the  centre  A, 
which  should  correspond  within  a  tenth  or  so  with  that  given 
on  the  location  profile.  By  subtracting  the  grade  at  the  station 
from  this  elevation  we  thus  have  H,  the  centre  cut  at  A.  The 
rodman  then  goes  to  the  left  and  holds  the  rod  at  some  point 
b  near  where  he  judges  the  slope-stake  will  come.  If  on  ob- 
taining the  side-height  for  b  it  is  found  that  the  proper  distance 
out  from  A  for  this  height  does  not  agree  with  the  distance 
out  as  actually  measured,  other  points  must  be  tried  until  a 
point  is  obtained,  such  as  B,  where  these  two  correspond.  An 
error  of  only  a  few  tenths  in  distance  can  be  estimated  for  by 
eye  without  taking  a  separate  reading  to  correct  for  it,  so  that 
two  or  three  trials  are  usually  all  that  are  required  to  fix  the 


RAILROAD   CONSTRUCTION. 


113* 


position  for  the  slope-stake  ;  and  on  comparatively  level 
ground  the  point  can  be  usually  hit  off  by  a  good  rodman  at 
the  first  trial. 

Similarly  on  the  right  the  point  C  must  be  fixed. 

If  there  are  any  decided  irregularities  in  the  surface,  such 
as  is  represented  at  D,  the  elevations  of  such  points  must  also 
be  taken. 

The  following  rules  give  all  that  is  required  as  regards  the 
actual  levelling : 

1.  When  H.I.  is  above  grade. — If  the  rod-reading  exceed 
the  difference  in  elevation  of  the  H.I.  and  Grade,  the  excess  =  the 
fill;  but  if  it  is  less,  the  deficiency  =  the  cut.    Consequently,  when 
the  rod-reading  =  the  difference  of  H.I.  and  Grade  that  point  is  a 
Grade-point. 

2.  When    H.I.  is    below    Grade,   the  rod-reading  -f-  the 
difference  of  H.I.  and  Grade  =  the  fill. 

Cut  is  always  indicated  by  a  positive,  and  Fill  by  a  negative 
sign. 
The  following  is  a  good  form  for  keeping  the  notes  : 


Sta. 

L. 

C. 

R. 

B.S. 

F.S. 

H.I. 

Elev. 

Grade. 

Re- 
marks. 

1020 

0.0 
~7i6 

+  1.0 

+  3.0 
14.5 

1.3 

10230 

101.0 

100.00 

$*ii 

1021 

-1.0 

0.0 

+  3.3  -f  1.0 

ITo"  iuf 

1.3 

101.0 

101.00 

ojf~ 

.  -3.0 

0.0 

^  c*  ~*    — 

1022 

11.5 

—  2.0 

7.0 

2.3 

100.0 

102.00 

+&m 

There  is  no  need  to  work  out  the  elevations  in  the  field,  but 
so  doing  in  the  office  afterwards  forms  a  useful  check  on  the 
work,  since  H.I.  —  F.S.  (which  of  course  is  the  elevation) 
should  agree  within  a  tenth  or  so  with'  the  sum  of  grade  ± 
centre-height,  F.S.  representing  the  rod-reading  at  the  centre. 
We  see  from  the  above  that  it  is  the  Difference  of  ILL  and  Grade 
which  is  the  foundation  of  the  calculation  at  each  station, 
and  this,  when  worked  out  for  the  next  station  after  a  turning- 
point,  can  be  modified  for  the  succeeding  stations  by  merely 
adding  or  subtracting  the  difference  in  grade.  Thus  the  cal- 
culation is  simpler  than  it  at  first  appears  from  the  above  rules. 

The  slope-stakes  should  be  marked  S.S.  on  the  outer  sides 


114  RAILROAD    CONSTRUCTION". 

and  the  numbers  of  the  stations  on  the  inner.     The  centre- 
stakes  should  have  the  cut  or  fill  marked  on  them. 
As  to  the   points  at  which  cross-sections  should  be 

taken,  the  rodmau  in  selecting  them  should  bear  in  mind  that 
it  is  not  necessarily  the  highest  or  lowest  points  that  are  re- 
quired, but  those  points  which,  when  joined  by  straight  lines, 
will  give  the  contents  as  nearly  as  possible  equal  to  the  true 
volume.  It  is  impossible  as  well  as  unnecessary  to  take  account 
of  many  of  the  small  irregularities  which  occur,  but  by  a 
judicious  selection  of  points  these  may  to  a  considerable  ex- 
tent be  made  to  counteract  each  other.  Where  the  contents 
are  calculated  by  "  average  areas" — as  is  usually  the  case — wre 
can  easily  find  from  Sec.  130  what  limit  should  be  adopted  as 
regards  the  difference  in  centre-heights  and  widths  between 
the  slope-stakes  of  two  cross-sections,  in  order  that  the  error 
in  the  volume  as  calculated  shall  not  exceed  a  certain  amount. 
For  exact  work  a  difference  of  two  feet  between  the  centre- 
heights  of  two  ad  joining  cross-sections  is  about  the  limit  which 
should  be  allowed  ;  but  in  ordinary  practice  we  may  say  that 
a  cross-section  should  be  taken  every  50  feet  when  the  differ- 
ence in  centre-height  amounts  to  about  5  feet.  This  is,  of 
course,  mainly  to  reduce  the  errors  which  arise  from  using 
an  approximate  method  of  calculating  the  quantities,  and  not 
to  take  into  consideration  the  irregularities  of  surface.  To 
counteract  as  much  as  possible  these  latter,  judgment  in  the 
selection  of  the  cross-sections  has  a  belter  effect  than  labor 
spent  in  obtaining  a  large  number  of  cross-sections  a  few  feet 
apart.  They  should  also  be  taken  whenever  "grade"  occurs 
on  either  the  edge  of  the  road-bed  or  in  the  centre  ;  and  when- 
ever a  cross-section  is  taken  where  a  grade-point  falls  in  the 
road-bed  its  position  must  be  obtained.  For  if  a  grade-point 
is  the  only  point  obtained  at  any  station,  it  necessitates  assum- 
ing centre-  and  side-heights  afterwards  in  working  out  the  con 
tents,  in  order  to  make  use  of  that  grade-point,  so  that  it  is 
much  more  satisfactory — and  in  the  end  involves  no  more 
work — to  obtain  these  heights  by  direct  measurement. 

There  is  of  course  no  need  to  take  cross-sections  any  closer 
together  on  a  curve  than  on  a  tangent,  as  may  be  easily  seen 
from  Sec.  134. 

When  in  doubt  as  to  the  material  in  a  certain  cut,  i.e.,  as  to 
whether  it  is  earth  or  rock,  etc.,  it  is  best  to  cross-section  it 


EAILROAD    CONSTRUCTION.  115 

for  the  usual  earth-slopes  and  have  it  stripped  to  that  width 
in  one  or  two  places  ;  if  then  rock  is  encountered  in  a  solid 
bed,  the  rest  of  the  cut  may  be  cross-sectioned  for  rock,  and 
as  soon  as  the  rock  is  reached  the  earth  trimmed  off  to  its 
proper  slopes  before  the  rock  is  worked.  This  of  course 
necessitates  a  cross-sectioning  of  the  rock  surface  as  well  as 
of  the  original  ground- surf  ace,  and  these  cross-sections  should 
be  taken  at  the  same  stations,  so  as  to  facilitate  the  calculation 
of  the  respective  volumes  of  earth  and  rock. 

107.  The  referencing  of  the  P.C.'s  and  P.T.'s  is  a  part  of 
the  engineer's  work  which  must  also  be  attended  to  before  con- 
struction begins.    Reference-points  should  be  placed,  two 
on  each  side  of  the  alignment,  at  angles  of  about  45°  with  it, 
and  sufficiently  distant  to  be  free  from  all  chance  of  disturb- 
ance during  construction  ;  the  point  referenced  thus  lies  at  the 
intersection  of  the  two   lines  joining    the   opposite    points. 
Sometimes,  however,  especially  on  side-hill  work,  it  is  neces- 
sary to  place  all  the  reference  points  on  one  side  of  the  track, 
in  which  case  the  apex  of  the  angle  formed  by  the  lines  pass- 
ing through  each  pair  of  reference-points  is  the  point  refer- 
enced.     Each  reference-point  should  be  marked  R.P.  on  a 
guard-stake  set  beside  it,  and  the  magnetic  bearings  and  dis- 
tances of  the  points  entered  in  the  notes. 

108.  The  Staking  out  of  Bor row-pits  consists  in  driving 
stakes  at  the  corners  of  the  proposed  pits,  and  obtaining  eleva- 
tions of  the  ground-surface  so  as  to  form  the  upper  lineof  a  set 
of  parallel  cross-sections  of  the  pit,  the  lower  line  being  ob- 
tained by  taking  levels  immediately  under  those  taken  on  the 
surface,  when  the  excavation  is  completed.     In  order  that  the 
bottom  levels  may  be   properly  connected  with  those  taken 
on  the  surface,  reference-points  must  be  established.      The 
simplest  way  of  doing  this  is  by  driving  hubs,  say  10  feet 
back  from  the  edge  of  the  pit,  in  the  line' of  each  cross-section. 
By  taking  the  cross-sections  27  feet  apart,  as  is  often  done, 
there  is  some  little  labor  saved  in  calculating  the  contents, 
since  the  mean  of  any  two  cross-sections  in  square  feet  equals 
the  volume  between  them  in  cubic  yards. 

A  sketch  plan  of  each  pit  should  be  made  in  the  note-book, 
and  properly  lettered  to  accord  with  the  notes. 

109.  Staking  out  Foundation-pits  for  Culverts,  either 
masonry  or  timber,  consists  of  setting  stakes  at  the  corners  as 


116 


RAILROAD    CONSTRUCTION. 


given  by  the  foundation  plan  and  marking  on  each  stake  the 
cut  necessary.  A  sketch  of  each  pit  should  be  made  in  the 
note-book,  and  of  course  the  amount  of  cut  at  each  stake  re- 
corded. When  the  foundation  consists  of  timber,  the  pit 
should  be  low  enough  to  insure  the  timber  being  at  all  times,  if 
possible,  kept  under  water,  or  at  any  rate  moist;  about  18  inches 
is  the  average  depth  for  foundation-pits  for  wooden  culverts 
on  Railroad  work.  In  staking  out,  it  should  also  be  remem- 
bered that  the  culverts  should  not  have  a  fall  of  more  than, 
say,  1  in  10,  so  that  when  the  ground  slopes  transversely  to  a 
greater  extent  than  this  the  culvert  must  be  put  on  the  skew 
so  that  its  inclination  "will  not  exceed  this  amount.  If  the 
depth  of  the  foundation-pit  exceeds  4  or  5  feet,  it  should  be 
staked  out  a  foot  wide  all  round  to  allow  room  for  working. 

110.  Setting  out  Bridge-foundations.— When  a  bridge 
is  on  a  tangent  there  is  no  difficulty  about  staking  out  the 
foundation-pits,  that  needs  particular  mention.  The  work  is 
usually  best  done  with  a  transit  and  tape  from  the  centre-line, 
— an  optical  square  comes  in  very  handy  for  this, — the  offsets 
being  obtained  by  scale  or  otherwise  from  the  foundation  plan. 
In  this  way  there  is  less  liability  to  make  an  error  than  in  any 
other,  since  each  point  is  set  out  independently  of  the  previous 
ones.  When  the  material  is  not  likely  to  stand  vertically,  it 
should  be  given  a  slope  sufficient  to  warrant  its  stability.  If 
there  is  not  room  to  admit  of  this,  then  of  course  the  sides 
must  be  shored-up  in  some  way, 

When,  however,  the  bridge  is  on  a  curve,  if  the  span  is 
short,  it  is  from  the  tangent  at 
the  centre  of  the  bridge  that 
the  offsets  must  be  set  off.  In 
dealing,  however,  with  bridges 
of  comparatively  long  spans, 
the  centre  of  the  curve  on  the 
bridge  will  by  no  means  coin- 
cide with  the  centre  of  the 
structure,  as  is  shown  by  Fig. 
56. 

Now  AB  will  be  the  centre- 
line of  the  bridge,  where 
cb  =  |  ordinate  at  M  to  ab 


— -'c 


y 

FIG.  56. 
(see  Equation  23,  Sec.  80);  so  that  the  true  centres  of  the  piers 


RAILROAD    CONSTRUCTION.  117 

lie  considerably  outside  the  centre-line  at  those  points.  If 
any  pier,  as  c,  is  inaccessible,  c  (its  centre)  may  be  located  as 
follows  : 

In  the  centre-line  of  the  track  take  some  accessible  point  P, 
and  set  off  PB  perpendicular  to  AB,  making 

PB  =  R  (vers  POM-  \  vers  bOM)  ; 
then  will 


then  be  located  either  by  direct  measurement  from 
B,  or  by  intersection. 

In  setting  out  bridge-foundations  great  care  should  be  given 
to  a  thorough  system  of  referencing  all  important  points, 
and  the  reference-points  must  be  so  selected  as  not  to  be  ob- 
structed by  staging  or  scaffolding  during  the  progress  of  the 
work. 

111.  Setting  out  Trestlework.  —  In  locating  the  position 
for  the  piles  in  low  pile-bents,  it  is  sufficient  to  locate  the 
centre  of  each  bent  and  then  set  off  the  positions  for  the  piles 
by  measuring  out  from  the  tangent  at  the  centre,  finding  the 
angle  by  eye  ;  if  possible,  the  position  of  each  pile  should  be 
marked  with  a  stake. 

When  piles  are  being  driven  on  a  curve  by  a  floating  pile- 
driver,  in  water  too  deep  to  drive  stakes,  the  centre  of  each 
bent  must  be  given  by  the  intersection  of-  the  lines  given  by 
two  transits,  as  in  Sec.  76. 

If,  however,  the  trestle  is  on  a  tangent,  by  placing  pickets 
on  either  bank  in  line  with  each  row  of  piles  the  centre  for 
any  pile  can  be  given  without  the  aid  of  an  instrument  ;  or 
pickets  can  be  so  set  that  the  pile-driver  can  line  itself  in 
without  the  assistance  of  any  one  on  the  bank  :  the  distances 
between  the  bents  may  be  taken  by  measurement  from  one 
bent  to  the  next.  In  the  case  of  framed  bents  resting  on  sills, 
it  is  advisable  to  have  the  sills  brought  to  a  solid  foundation 
at  about  an  indicated  elevation  before  the  framing-bill  is  made 
out  :  in  this  way  a  firmer  foundation  is  often  obtained  at  a 
cost  of  less  labor  than  if  the  exact  elevation  for  the  sills  was 
prescribed.  The  sills  for  each  "bent  should  then  be  accurately 
levelled  and  centred. 

In  dealing  with  high  trestles,  the  transverse  centre-line  of 


118  RAILROAD   CONSTRUCTION. 

each  betit  should  be  referenced,  the  reference-points  being  at 
a  considerable  distance  from  the  bent  itself,  so  as  the  better 
to  permit  the  line  being  carried  to  a  high  elevation  in  the 
structure  if  required.  The  length  of  the  chords  should  be 
corrected  according  to  Sec.  76. 

Where  pony-bents  are  used  they  should  be  so  skewed 
around  as  to  conform  with  the  contour  of  the  ground  ;  they 
must  be  accurately  levelled  before  the  sills  are  laid  on. 

In  giving  points  for  "cut-off's"  in  piling  out  of  reach,  the 
pile  should  be  blazed  and  a  tack  driven  into  it,  the  distance 
above  the  tack — which  should  be  in  full  feet — being  inscribed. 
The  position  of  the  tack  is  best  found  as  follows :  For  ex- 
ample, let  the  difference  of  H.I.  and  grade  =  6.11  feet ;  then 
if  the  pomt  of  cut-off  is  2  feet  below  grade,  and  it  is  wished 
to  put  in  the  tack  so  as  to  read  "5  feet  below  cut-off,"  we  must 
read  on  the  rod  0.89  foot.  The  position  of  the  tack  is  then 
at  the  foot  of  the  rod. 

112.  Setting  out  Tunnels.— This  is  work  which  often 
needs  considerable  time  and  care,  in  order  that  the  results  ob- 
tained may  be  satisfactory. 

Let  Fig.  57  represent  the  section  of  a  tunnel  in  course  of 
construction. 


FIG.  57. 


The  first  thing  to  do  is  to  establish  some  point  C  in  the 
alignment  from  which  a  good  view—  if  possible—  may  be  had 
of  the  mouths  of  any  shafts  which  it  may  be  required  to  sink, 
and  also  of  two  distant  points  A  and  B,  also  in  the  same 
straight  line.  If  the  instrument  is  then  set  up  at  C  and  the 
telescope  clamped  on  to  A,  on  reversing  it  the  point  B  should 
be  intersected.  By  repeated  trials  the  three  points  A,  B,  and 
C  are  then  established  in  the  same  straight  line,  and  these 
points  should  be  permanently  marked. 

In  order  to  obtain  the  centre-line  of  the  tunnel,  say  at  the 
left  end,  another  point  G  in  the  same  line  as  AB  must  be 


RAILROAD   CONSTRUCTION. 


119 


given,  and  the  centre-line  is  then  obtained  by  the  production 
of  AG. 

But  suppose  the  work  is  to  be  carried  on  also  from  one  or 
more  shafts  as  EF,  then  the  alignment  has  to  be  "dropped" 
from  ED  to  the  elevation  of  the  tunnel  at  F,  and  in  this 
operation  the  greatest  care  is  necessary.  There  are  three  or 
four  ways  in  which  this  can  be  done,  but  the  following  is  that 
usually  adopted  for  tunnel-work,  as  it  admits  of  greater  ac- 
curacy than  the  others,  which  are  more  suitable  for  simpler 
mining  operations  : 

Two  instruments  such  as  that  shown  in  Fig.  58  should  be 
firmly  bolted  on  either  side  of  the  shaft  as  D  and  E,  and  near 
to  its  edge,  both  being  lined  in  vertically  over  the  centre-line 
of  the  tunnel. 

Each  instrument  consists  of  a  plate  p — with  a  narrow  verti- 
cal slit  in  it  and  scale  s  attached — which  can  be  moved  side- 
ways by  means  of  the  screws  a  and  b,  so  that  it  can  be  set  to 
any  desired  reading  on  the  scale— 
the  scale  being  read  by  a  vernier 
v  attached  to  the  main  body  of  the 
instrument.  Having  set  these  two 
instruments  approximately  in  line, 
then,  by  a  series  of  observations 
taken  at  different  times, — so  as  to 
counteract  as  much  as  possible  the 
varying  conditions  which  affect 
each  separate  sight,— ascertain  for 
each  instrument  the  mean  of  the  FlG-  58- 

readings.     Having  then  set  the  plates  to  give  that  reading,  the 
centres  of  the  vertical  slits  coincide  with  the  mean  alignment. 

Two  fine  steel  wires  must  then  be  carried  from  one  slit  to 
the  other,  each  being  placed  against  the  vertical  edge,  so  that 
they  form  two  parallel  lines,  close  together,  across  the  shaft, 
one  on  each  side  of  the  alignment.  Midway  between  these 
two  wires,  and  as  near  to  the  edge  of  the  shaft  as  possible,  but 
on  opposite  sides  .of  it,  two  fine  copper  wires  should  be  passed, 
long  enough  to  reach  down  to  the  tunnel  at  F,  and  to  the  ends 
of  these  two  heavy  plumb-bobs  should  be  attached.  The  wires 
should  be  enclosed  in  wooden  tubes  to  protect  them  from  cur- 
rents of  air,  falling  water,  etc.  The  plumb-bobs  themselves 
should  be  immersed  in  buckets  of  water  to  lessen  their  oscilla- 


120  RAILROAD   CONSTRUCTION. 

tions.  Scales  should  then  be  placed  so  as  to  read  these  oscil- 
lations slightly  above  the  plumb-bobs.  The  mean  of  these 
sets  of  readings  then  gives  a  point  on  the  alignment,  and  from 
the  two  points  so  obtained  the  centre-line  of  the  tunnel  may 
be  extended  in  either  direction  by  first  establishing  a  point  in 
one  direction,  and  then  in  the  other ;  and  these  points  can  then 
be  checked  by  observing  whether  all  four  are  in  the  same 
straight  line :  if  found  to  be  correct,  they  should  be  perma- 
nently established.  The  levels  may  be  dropped  by  means  of  a 
steel  tape,  with  which  the  levelling-rod  used  has  been  pre- 
viously compared. 

The  length  of  the  tunnel  may  be  found  either  by  direct 
measurement  (breaking-chain)  or  by  triangulating. 

In  locating  a  tunnel,  it  should  be  remembered  that  it  is 
usually  cheaper  to  open  a  cut  at  depths  under  60  feet  than  to 
bore.  In  many  clays,  however,  a  cut  of  this  depth  would  be 
barely  practicable  owing  to  the  increase  in  the  inclination  of 
the  slopes  necessary  on  account  of  the  depth  itself,  and  in  such 
cases  the  limit  is  considerably  less  than  this.  As  regards  the 
advisability  of  sinking  shafts,  it  is  mainly  a  question  of  the 
depth  of  shaft  required,  the  need  of  ventilation,  and  the 
facilitating  the  transport  of  material.  Where  the  depth  is  not 
excessive  it  is  usually  policy  to  sink  several  shafts  in  a  long 
tunnel,  and  work  from  each'  independently,  for  the  work  is 
thereby  considerably  hastened,  and  after  its  completion  the 
shafts  themselves  form  admirable  means  of  ventilation. 

Side-drifts,  where  they  are  possible,  accomplish  the  same  re- 
sults as  shafts,  and  are  usually  to  be  preferred  to  them  on  ac- 
count of  less  risk  to  life  and  property  during  construction, 
and  their  convenience  afterwards. 

Where  the  alignment  has  not  to  be  carried  to  any  great  dis- 
tance from  the  points  dropped  to  the  bottom  of  a  shaft  as 
a oove  described,  it  is  better  to  sink  the  shaft  a  few  feet  on  one 
side  of  the  centre-line,  and  to  reach  the  tunnel  from  it  by 
means  of  a  cross-heading. 

The  centre  line  in  the  tunnel  is  best  given  by  points  on  the 
rooi  from  which  plumb-lines  can  be  hung  when  required. 

113  Giving  Grade  and  Centres  forms  a  very  large  por- 
tion of  the  work  to  be  done  by  the  engineer  during  construc- 
tion. The  giving  of  "grade"  may  be  greatly  facilitated  by 
having  stakes  driven  to  grade,  from  which  at  any  future  time 


RAILROAD    CONSTRUCTION.  121 

the  levels  may  be  given  with  a  hand-level  -an  instrument 
highly  useful  during  railroad  construction.  To  have  to  carry 
a  heavy  level  for  several  miles  just  to  give  grade  at  two  or 
three  stations,  as  is  frequently  done,  is  absurd.  By  having  a 
bubble-tube  attached  to  the  telescope  of  the  transit  a  consider- 
able amount  of  trouble  may  also  be  saved,  and  with  it  the 
elevations  can  be  given  quite  as  correctly  as  are  ever  required 
on  a  railroad  dump. 

In  setting  grade-stakes,  allowance  must  be  made  in  dealing' 
with  material  which  is  likely  to  shrink  in  order  to  allow  for  it. 
The  amount  of  the  Shrinkage  depends  considerably  on  the 
pressure  to  which  the  material  is  subjected,  consequently  on 
the  height  of  the  fill :  as  an  average,  however,  in  earthy  soils 
the  linear  contraction  is  about  10  p.  c.,  so  that  a  10-foot  fill 
should  be  "  put  up"  1  foot  above  grade.  In  dealing  with  wet 
or  frozen  soils  greater  allowance  should  be  made,  but  with 
dry  sandy  material,  less. 

The  allowance  also  depends  very  largely  on  the  manner  in 
which  the  dump  is  constructed.  A  dump  well  trodden  by 
horses  usually  shrinks  very  little,  and  in  many  such  cases 
there  is  no  need  to  allow  for  shrinkage  at  all ;  but  where  the 
work  is  put  up  by  tipping  or  shovelling,  double  the  allowance 
may  in  some  cases  be  none  too  much. 

The  increase  in  bulk  in  roftc,  as  well  as  the  shrinkage  of 
earth,  necessitates  an  allowance  being  made  when  arranging 
for  the  distribution  of  material.  A  good  general  rule  for  this 
is,  that  10  yards  of  earth  in  excavation  make  9  yards  in  em- 
bankment, and  10  yards  of  rock  in  excavation  make  17  yards 
in  embankment. 

As  regards  "  giving  centres"  during  construction,  it  should 
be  seen  that  the  slope-stakes  are  intact,  and  then  by  their 
means  the  centres  for  a  cut  or  fill  may  be  usually  obtained 
from  the  cross-section  notes,  without  the  trouble  of  setting  up 
the  transit,  with  accuracy  quite  sufficient  to  enable  the  con- 
tractor to  proceed  with  his  work. 

114.  Difference  of  Elevation  on  Curves.— The  centrif- 
ugal force  brought  into  play  by  the  inertia  of  the  train  when 
going  round  a  curve  must  be  counterbalanced  by  a  more  or 
less  equal  and  opposite  force  in  order  to  prevent  the  flanges  of 
the  outer  wheels  being  pressed  too  severely  against  the  rails. 
The  simplest  way  of  bringing  a  counteracting  force  into  play 


122  RAILROAD   CONSTRUCTION. 

is  to  make  use  of  a  component  of  the  weight  itself,  which 

Centrifugal        mav  be  done  bv  canting 
*"  the  track  as  in  Fig.  59. 

Thus,  if  the  force  W, 
representing  the  weight 
of  a  car,  be  resolved  into 
its  rectangular  compo- 
nents JV  (normal  to  the 
track)  and  F  (parallel  to 
the  track),  we  see  from 
Sec.  7  that  F  is  propor- 

TT 

tional  to  —  ,  H  being  the  difference  in  elevation  of  the  rails, 

and    O   the   gauge  —  or    more   strictly,    the   distance    from 
centre  to  centre  of   rails.     Now  the  value  of  the  centrif- 

ugal force  in  pounds  equals  -^,  where  v  =  velocity  in  feet 


per  second,  and  R  the  radius  of  the  curve  ;  so  that  when 
there  is  no  tendency  to  tip  over  on  either  side  —  if  we  as- 
sume, as  we  may  well  do  in  practice,  that  F  is  the  com- 
ponent parallel  to  the  centrifugal  force  —  we  have 

H       tf  Gtf 

=     >  the*fore  H 


So  that,  substituting  for  R  the  value  given  in  Sec.  71,  and 
substituting  V,  velocity  in  miles  per  hour,  for  v,  we  have 

H=  .00067  #F2  sin  D  ', 
or,  as  an  approximate  formula,  easy  to  remember,  we  have 


If  we  take  G  =  4'  8£",  we  then  have 

#=.0032  F2sinD. 

The  following  table,  abbreviated  from  that  given  by  Mr. 
Searles,  calculated  for  the  value  of  F  parallel  to  the  centrifu. 


RAILROAD    CONSTRUCTION. 


123 


gal  force,  and  for  a  distance  from  centre  to  centre  of  rail 
=  4'  10|"  (suitable  to  the  4'  8V  gauge),  gives  the  difference  in 
elevation  of  the  two  rails  in  feet,  at  various  speeds  for  different 
degrees  of  curvature. 


Vel.  in 
m.  p.  h. 

DEGREE  OF  CURVE. 

1 

2° 

3° 

4° 

5° 

6° 

7° 

9° 

12° 

16° 

10 

.006 

.011 

.017 

.023 

.029 

.034 

.040 

.051 

.069 

.091 

20 

.023 

.046 

.069 

.091 

.114 

.137 

.160 

.20(3 

.274 

.365 

30 

.051 

.103 

.154 

.206 

.257 

.308 

.359 

.460 

.611 

.809 

40 

.091 

.183 

.274 

.365 

.455 

.545 

.634 

811 

1.069 



50 

.143 

.285 

.427 

.568 

.707 

.844 

.979 

— 

— 



60 

.206 

.410 

.612 

.811 

1.006 

1.196 

— 

— 

— 

— 

A  convenient  rule,  much  used  in  practice  for  a  gauge  of 
4'  8|",  is,  that  the  difference  in  elevation  equals  one  half  inch 
for  every  degree  of  curvature. 

In  order  to  allow  for  the  difference  in  elevation  on  the  dump, 
the  road-bed  should  have  its  outer  edge  higher,  and  its  inner 
edge  lower,  than  grade.  To  allow  for  it  on  trestles,  whether 
in  pile-bents  or  framed  bents,  the  posts  must  be  cut  so  as  to 
give  the  required  inclination  to  the  cap  on  which  the  stringers 
rest :  the  batter  of  the  batter-posts  and  the  verticality  of  the 
upright  posts  remain  unchanged. 

It  is  usual  to  adopt  a  difference  in  elevation  in  the  rails 
suitable  to  the  mean  speed  of  the  trains  which  pass  over  them  : 
the  consequence  of  which  is,  that  the  rails  on  both  sides  get 
worn,  but  in  different  ways — the  outer  ones  by  the  fast  trains 
and  the  inner  ones  by  the  slow  trains.  The  coning  of  wheels, 
which  was  at  one  time  largely  resorted  to,  is  rarely  used  now 
on  account  of  the  increased  oscillation  and  concussion  (see 
Sec.  4)  to  which  it  gave  rise,  so  that  the  flanges  of  the  wheels, 
by  means  of  their  pressure  against  the  inner  sides  of  the  rails, 
have  themselves  to  keep  the  balance  between  the  centrifugal 
force  and  the  component  of  gravity  which  is  set  to  counteract 
it,  more  or  less.  In  curves  uneased  by  transition  curves,  the 
difference  in  elevation  at  the  P.O.  and  P.T.  must  be  at  least 
equal  to  what  it  is  at  any  other  Dart  of  the  curve,  so  that  it 
must  begin  some  little  distance  back  on  the  tangent  and  in- 
crease gradually  until  it  reaches  its  maximum  at  the  P.O.  or 


124  RAILROAD   CONSTRUCTION. 

P.T. ,  as  the  case  may  be.  For  a  3°  curve  it  isNisually  sufficient 
to  begin  the  difference  in  elevation  about  100  feet  back,  aud 
for  a  10°  curve  about  200  feet  back  on  the  tangent.  When 
transition  curves  are  used,  they  must  be  treated  with  a  differ- 
ence in  elevation  at  all  points  more  or  less  suitable  to  their 
curvature  ;  but  where  the  transition  curve  is  merely  a  simple 
curve  inserted  to  ease  the  approach  to  a  sharper  one,  the 
difference  in  elevation  for  the  terminal  curve  must  begin  back 
on  the  tangent  as  above,  and  for  the  main  curve  some  little 
distance  back  on  the  terminal  curve,  so  as  to  admit  of  its 
reaching  its  maximum  at  the  P.C.C. 

It  is  usual  to  slightly  increase  the  gauge  on  curves,  generally 
by  about  i"  for  every  degree  of  curvature  up  to  5°. 

115.  Inspecting  the  Grading. — The  engineer  should,  if  pos- 
sible, pass  over  every  portion  of  his  subdivision  at  least  twice  a 
week,  and  the  oftener  the  better.  In  open  country  there  is 
comparatively  little  chance  of  having  the  dump  badly  put  up 
owing  to  lack  of  supervision,  except  perhaps  through  the  use 
of  a  superabundance  of  "  sods  ;"  but  in  timber  country  where 
there  is  plenty  of  grubbing  to  be  done,  and  the  work  is 
largely  let  as  "station-work,"  the  engineer  must  be  constantly 
on  the  lookout  for  the  presence  of  roots  and  stumps  in  the 
dump.  In  winter  too,  snow,  frozen  moss,  etc.,  at  the  bottom 
of  a  fill  serve  admirably  as  a  temporary  means  of  bringing  it 
up  to  grade.  He  should  see  that  there  is  a  fair  line  of  stumps 
at  the  side  of  the  track  after  the  completion  of  the  work  in 
places  where  grubbing  has  occurred,  or  that  they  have  really 
been  burnt  ;  and  when  there  is  snow  on  the  ground  he  must 
have  it  swept  well  to  the  side  before  the  filling  is  begun.  He 
must  see  that  the  ditches  on  either  side  of  the  embankments, 
etc.,  as  well  as  those  in  the  cuts  themselves,  are  taken  out 
properly,  aud  thoroughly  cleared  of  all  obstructions,  that  the 
slopes  are  neatly  dressed  off  and  well  out  to  the  slope-stakes. 
For  the  final  inspection  of  the  road-bed,  grades  and  centres 
must  be  carefully  run,  and  the  width  tested  wherever  it  ap- 
pears lacking.  All  litter  along  the  side  of  the  track  must  be 
cleared  away  or  burnt,  and  anything  in  danger  of  falling  on 
to  the  road-bed  removed.  About  this  latter  injunction  the  er  • 
gineer  cannot  be  too  careful,  and  when  in  doubt  as  to  the  stabil- 
ity of  a  piece  of  rock  or  an  overhanging  tree,  he  should  have  it 
removed  at  any  cost.  He  must  also  remember  that  a  rock  or 


RAILROAD    CONSTRUCTION.  125 

tree  which  at  the  time  of  inspection  looks  tolerably  firm, 
may  be  a  considerable  source  of  danger  after  the  disintegrat- 
ing effects  of  a  hard  winter,  or  a  season  of  heavy  rains,  and 
that  it  costs  very  much  less  to  have  it  removed  during  con- 
struction than  at  a  later  period. 

110.  Running  Track-centres  and  setting  Ballast-stakes. — 
Where  the  ballasting  is  done  before  the  track  is  laid,  ballast- 
stakes  must  be  driven  every  50  feet,  so  that  their  tops  indicate 
the  elevation  of  the  top  of  the  ballast.  They  should  be 
placed  on  either  side  of  the  centre-line  at  the  foot  of  the 
ballast-slopes.  Centre-stakes  should  also  be  set  every  100  feet 
apart  on  tangents  and  every  50  feet  apart  on  curves,  to  guide 
the  track-layers  ;  tacks  should  be  inserted  in  them. 

When  the  track  is  laid  without  first  ballasting,  a  line  of  cen- 
tres must  be  given  before  the  track  is  laid,  and  usually  after- 
wards as  well,  to  guide  the  surfacing  gang,  lor  the  centres 
previously  put  in  are  almost  sure  to  have  been  knocked  out  in 
laying  the  track. 

It  sometimes  happens  in  hasty  work  that  the  engineer  who 
has  the  track-centres  to  run  cannot  get  his  centres  to  coincide 
with  the  centre  of  the  dump  or  with  the  centres  of  the 
bridges.  As  regards  the  centres  on  the  dump,  he  must  use  his 
own  judgment  as  to  what  is  best  to  do  :  if  it  is  clear  that  the 
dump  is  out  of  line,  he  must  stand  by  his  own  centres  ;  but  if 
otherwise,  it  is  usually  better  for  him  to  increase  or  ease  his 
curvature  a  little,  so  as  to  make  it  conform  with  the  centre  of 
the  road-bed.  On  bridges  or  open  culverts  lie  must  make  his 
own  centres  fit  the  centres  of  the  structures,  and  if  this  can- 
not be  done  without  seriously  affecting  the  adjacent  track,  the 
case  must  be  reported  at  once. 

117.  Permanent  Reference-points. — After  the  track  is  laid, 
large  hardwood  stakes — or  better  still,  stone  monuments — 
should  be  set  to  mark  the  P.C.'s,  P.C.€.'s,  and  P.T.'s.  They 
should  be  placed  on  the  outer  side  of  the  curves,  at  right 
angles  to  the  track,  usually  about  5  or  6  feet  from  the  centre. 


TURNOUTS  AND  CROSSINGS. 

118.  In  dealing  with  the  subject  of  turnouts  and  crossings, 
we  will  assume  that  the  Common  Stub  Switch  is  used,  since  it 


126 


RAILROAD   CONSTRUCTION. 


is  the  simplest,  and  the  formulae  for  it  are  readily  applied  to 
any  other  form  of  switch. 

Let  Fig.  60  represent  a  turnout  from  a  straight  track,  A  and 
a  forming  the  "  heel  "  and  B  and  b  the  "  toe"  of  the  switch. 


FIG.  60. 


Then  if 
O  =  gauge, 

2f=  number  of  the  frog, 
F—  "Frog  angle," 
=  Angle  of  Intersection 
at  F, 


we  have 


cot 


H  —  radius  of  turnout  curve, 
AF  =  frog  distance, 
AB  —  length  of  switch-rail, 

D  =  degree  of  curve, 


G 


AF  =  ZGN, 


R  = 


AF=(ll+®}  sin  F, 

'}-- 

I       8 


AB  = 


X  Throw. 


The  throw  according  to  Sec.  78  =  -^=-  • 

att 

The  number  of  a  frog  may  of  course  always  be  found  by 
measuring  the  tongue:  thus  if  at  a  certain  point  we  find  its 
width  to  be  5  inches,  this  divided  into  the  distance  of  that 
point  from  the  theoretic  point  of  the  tongue  gives  the  number 
of  the  frog;  thus  if  that  distance  were  4'  2",  it  would  be  a 
No.  10  frog. 


RAILROAD   CONSTRUCTION. 


127 


The  following  table  gives  these  values  for  a  gauge  of  4' 
and  a  throw  of  5". 


N 

F 

AF  in  feet. 

R  in  feet. 

D 

AB  in  ft. 

4 

14°  15' 

37.66 

150.66 

38°  46' 

11.2 

5 

11°  25' 

47.08 

235.40 

24°  32' 

14.0 

6 

9°  32' 

56.50 

338.98 

16°  58' 

16.8 

7 

8°  10' 

65.91 

461.38 

12°  27' 

19.6 

8 

7°  09' 

75.33 

602.62 

9°  31' 

22.4 

9 

6°  22' 

84.74 

762.70 

7°  31' 

25.2 

10 

5°  43' 

94.16 

941.60 

6°  05' 

28.0 

11 

5°  12' 

103.58 

1139.34 

5°  02' 

30.8 

12 

4°  46' 

112.99 

1355.90 

4°  14' 

33.6 

This  table  may  be  applied  to  other  gauges;  F  of  course  re- 
maining unchanged,  AF  and  R  will  vary  directly  as  the 
gauge  ;  D  will,  of  course,  vary  inversely  as  R.  Thus  for  a 
3-foot  gauge  and  a  No.  9  Frog  we  must  multiply  the  above 

values  of  AF  and  R  by    '        =  .637 ;  and  the  above  value  of 

D  must  be  multiplied  by    '        =1.57.    AB  is  of  course  de- 
o 

pendent  on  the  value  of  the  throw  adopted. 

119,  Suppose,  however,  that  the  turnout  instead  of  starting 
from  a  straight  track,  as  in  Fig.  60,  starts  from  a  curve  as  in 
Figs.  61  and  62 ;  then  we  may  assume  that  when  the  main 
curve  and  the  turnout  curve  are  both  in  the  same  direction, 
that  the  case,  as  regards  the  position  of  the  frog,  etc.,  is  equiva- 
lent to  a  turnout  from  a  straight  track,  the  curvature  of  the 
turnout  curve  being  equal  to  the  difference  of  the  curvature  of 
the  main  and  of  the  turnout  curve;  and  if  in.  opposite  directions, 
then  the  curvature  of  the  turnout  curve  may  be  taken  as  being 
equal  to  the  sum  of  the  curvatures. 


FIG.  61.  FIG.  62. 

Suppose  we  have  two  parallel  tracks  AD  and  CB,  as  in  Fig. 
63,  which  we  wish  to  join  by  a  crossing;  or,  having  the  track 
AD  only,  we  wish  to  insert  a  turnout  AB  which  shall  connect 
the  side  track  B  with  the  main  track  AD.  Since  the  former 
case  differs  only  from  the  latter  in  the  fact  that  the  dotted 


128  RAILROAD   CONSTRUCTION. 

portion  C,  with  the  accompanying  frog,  is  omitted,  the  two 
cases  may  be  treated  together  as  follows: 


A-J 


FIG.  63.  £ 

Starting  from  the  centre-line  .^.j^with  a  given  frog  number, 
we  select  a  certain  length  n,  expressing  the  length  of  the 
branch  AM  in  terms  of  100-foot  stations.  The  length  of  the 
offset  t  at  M  is  then  given,  according  to  Sec.  78,  by  the  formula 

t  =  R  vers  nD, 
and  the  distance  along  the  track  AD  to  this  offset  equals 


Thus  by  setting  off  the  offset  t  at  a  distance  T  along  the 
tangent  from  A,  we  locate  the  point  M.  The  position  of  the 
frog  at  Fis  found  by  taking  from  the  above  table  the  value  of 
AF,  and  measuring  it  off  along  AD,  offsetting  F  by  an  amount 
equal  to  half  the  gauge. 

Another  offset  y  =  £  gauge  may  also  be  set  off  at  a  tangential 
distance  =  $AF.  These  points,  together  with  the  toe  of  the 
switch,  are  usually  all  that  are  wanted  in  the  curve  AM.  The 
length  of  any  other  offset,  if  required,  may  be  found  from 
Sec.  78. 

The  offset  t  is  then  produced  across  to  the  centre  of  the 
other  track  (or  the  other  track  produced)  and—  assuming  both 
branches  to  have  the  same  radius  —  the  offset  JVe  =  tis  set  off 
from  the  point  e,  which  point  is  found  from  the  formula 
ce  —  (d  —  2t)  cot  nD. 

We  thus  have  the  point  N.  The  curve  NB  is  then  located  by 
using  the  same  value  of  T,  and  the  same  offsets  as  before,  only 
of  course  in  reverse  order. 

By  obtaining  n  from  the  formula 

vers  nD  —  -=r=^ 

&H 

which  gives  its  limiting  value,  we  have  a  simple  reverse  curve 


RAILROAD   CONSTRUCTION.  129 

without  the  intervening  tangent  MN :  but  this  is  bad  practice 
when  it  can  be  avoided. 

Should  the  radius  of  NB  be  required  different  from  that  of 
AM,  the  tangential  distance  for  NB  must  then  be  calculated 
afresh. 

The  advantages  of  this  method  are,  that  any  length  of  inter- 
vening tangent  can  be  used, — provided  that  the  curves  are 
carried  up  to  the  frogs,— so  that  the  engineer  can  select  any 
value  of  n  for  himself;  and  with  simply  a  tape,  he  can  locate 
the  crossing  in  a  manner  a  good  deal  simpler  than  the  ways 
ordinarily  in  use. 

120.  As  an  example,  let  d  =  40  feet  and  let  No.  8  frogs  be 
used;  and  suppose  we  select  1.3  as  a  value  for  n.  Then  from 
the  table,  AF  =  75.33,  R  =  602.62,  and  D  =  9°  31',— the  gauge 
being  4'  8i". 

Then  from  the  above  formulae  we  have 

t  =  602.6  X  vers  12°  22'  =  14  feet, 
T—  602.6  X  sin  12°  22  =  129  feet, 
ce  =  12  x  cot  12°  22'  =  54.7  feet, 
and#  =  1.2  feet. 

The  notes  for  the  setting  out  of  the  crossing  may  then  be 
arranged  as  follows: 

8    3          3£ 


6 
FIG.  64. 

When  the  distance  between  the  two  tracks  is  great,  the  cross- 
ing should  be  run  in  with  a  transit. 

121.  If  the  turnout  or  crossing  falls  on  a  curve,  it  is  best  to 
locate  it  with  a  transit  according  to  one  of  the  two  following 
methods: 

1.  If  the  curvature  of  the  main  track  is  tolerably  sharp  and 
the  distance  d  between  the  centres  of  the  two  parallel  tracks 
comparatively  small,  we  can  avoid  the  insertion  of  a  reverse 
curve  without  materially  lengthening  the  crossing  as  follows: 


130  RAILROAD   CONSTRUCTION-. 

In  Fig.  65  let  D  =  the  degree  of  the  turnout  curve  AC, 

H  =  radius  of  the  outer  track  A, 
and  r  =  radius  of  the  turnout  curve  AC 
The  length  of  AC  may  then  be  found  in  terms  of  nD,  thus: 


and  the  length  of  the  tangent  equals 

CB=(R-  r)  sin  nD. 

For  example,  let  the  outer  track  A  be  on  a  4°  curve;  then 
R  =  1433,  and  let  d  =  40  feet,  and  the  given  frog  number  for 
the  main  curve  =  11. 

Then,  according  to  Sec.  119,  D  for  the  turnout  curve  must 
be  that  value  which  is  required  to  make  the  difference  in 
curvature  of  the  track  A  and  the  curve  AC  equal  about  5°, 
both  curves  being  in  the  same  direction;  and  since  this  value 


FIQ.  65. 

is  9°,  therefore  r  =  637  feet.  Set  the  instrument  up  at  A  and 
locate  the  9°  curve  AC;  and  since  by  the  above  formula 
nD  =  18°  15',  therefore  the  length  of  A  C  =202.7  feet,  and 
similarly  the  length  of  CB  =  249.2  feet.  Thus  we  find  the 
point  B. 

To  run  from  B  to  A  would  be  simply  a  reversal  of  the  above. 

The  frog  for  the  track  B  will  of  course  be  that  suitable  to  a 
turnout  radius  equal  to  the  radius  of  the  track  B. 

But  suppose  this  method  would  in  any  particular  case  cover 
too  much  ground,  or  be  unsuitable  in  some  other  respect,  we 
can  then  use  the  following  one,  which,  though  involving  the 
use  of  a  reverse  curve,  is  well  enough  for  station-yards,  etc., 
where  no  high  speeds  are  attained. 


RAILROAD   CONSTRUCTION.  131 

2.  In  Fig.  66  let  R  =  radius  of  the  inner  track  B, 
r  =  radius  of  branch  CB, 
TI  =  radius  of  branch  AG. 
Then 


vers  BEG  = 
from  which  we  can  find  the  length  of  the  branch  BG\  and 


vers  BOA  = 

and  since  the  angle 

AEG  =  BO  A -\-BHG, 

we  can  thus  find  the  length  of  the  arc  AG,  and  locate  the 
crossing  with  the  transit,  starting  from  either  end  A  or  B. 


FIG.  66. 

In  order  to  use  frogs  of  the  same  number  for  tracks  A  and  B, 
we  must  have 

r  =  7*1  -j-  d  (nearly). 

The  positions  of  the  frogs  maybe  found  according  to  Sec.  119. 
In  the  case  of  a  Double  Turnout  the  engineer  can,  by  ap- 
plying the  formulae  given  above,  always  locate  it  with  ac- 
curacy sufficient  for  ordinary  purposes,  without  the  aid  of 
special  formulae.  The  length  of  switch-rails  given  in  Table  in 
Sec.  118  are  the  proper  lengths  for  a  5"  throw,  but  in  practice 
a  difference  of  5  feet  or  so  in  the  length  of  the  rail  will  be  of 
very  little  importance.  In  the  same  way  there  is  no  necessity 
for  the  frog  to  have  exactly  the  number  which  it  should  have 
according  to  the  table.  The  laxity  which  is  allowable  in  these 
matters  depends  on  the  speeds  at  which  the  trains  are  likely  to 
pass  over  the  switch. 


132 


RAILROAD   CONSTRUCTION". 


122.  Curving  Rails.— The  following  table  gives  the  mid- 
ordinates  in  inches  for  curves  of  various  lengths.  Rails  should 
also  be  tested  for  Uniformity  of  Curvature  by  testing  one  half 
of  their  length  for  £  of  the  inid-ordiuate.  (See  Sec.  80.) 


LENGTH  OP  RAILS  IN  FEET. 

DKG.  OF 

30 

28 

26 

20 

18 

14 

10 

CURVE 

In. 

In. 

In. 

In. 

In. 

In. 

In. 

1° 

240 

.192 

.156 

.096 

.072 

.048 

.024 

2° 

.456 

.408 

.348 

.204 

.168 

.096 

.048 

3° 

696 

.612 

.528 

.312 

.264 

.144 

.072 

4° 

.948 

.828 

.720 

.420 

.348 

.216 

.108 

5° 

1  19 

1.03 

.888 

.528 

.420 

.264 

.132 

6° 

1  40 

1.22 

1.06 

.624 

.504 

.312 

.156 

1  64 

1.44 

1.25 

.732 

.588 

.360 

.180 

QO 

1  90 

1.64 

1.43 

.840 

.672 

.408 

.204 

10° 

2  35 

2.05 

1.78 

1.04 

.852 

.540 

.264 

12° 

2.83 

2.47 

2.15 

1.26 

1.02 

.636 

.312 

14° 

3  30 

2  87 

2.48 

1.46 

1.19 

.732 

.360 

16° 

3.76 

3.28 

2.83 

1.67 

1.36 

.840 

.420 

123.  Expansion  of  Kails.— Steel  expands  about  1  part  in 
150,000  for  each  degree  Fah.  through  which  its  temperature  is 
raised;  so  that  for  30-ft.  rails  the  spaces  between  their  ends 
should  vary  from  about  Ty  at  a  temperature  of  120°  F.  to 
about  TV '  at  a  temperature  of  -  40°  F.  This  must  be  carefully 
attended  to. 


B.  THE  ESTIMATING  OF  LABOR  AND  MATERIAL. 

124.  The  Expense  of  Grading  is  of  course  almost  entirely 
dependent  on  the  cost  of  the  labor  expended  on  it,  the  value 
of  the  material  not  entering  into  the  question  ;  so  that  esti- 
mating the  cost  of  it  is  simply  a  matter  of  ascertaining  the 
time  and  wages  which  are  absorbed  in  its  execution. 

The  following  notes  on  the  subject  of  handling  earth  and 
rock,  which  are  taken  from  Trautwine  on  Excavations  and  Em- 
bankments,— than  whom  possibly  no  better  authority  could 
be  quoted, — serve  to  show  the  relative  cost  of  the  different 
processes  through  which  the  material  has  to  pass  before  being 
finally  disposed  of  in  the  embankment;  and,  consequently, 
from  them  the  aggregate  cost  may  be  obtained  with  a  greater  or 
less  amount  of  precision.  These  processes  we  will  consider 
in  the  order  in  which  they  occur,  taking  as  the  standard  of 


RAILROAD   CONSTRUCTION.  133 

wages  $1.00  per  working  day  of  10  hours,  and  the  expense  of  a 
horse  as  $0.75  (including  Sundays). 

A.    THE   COST   OF   EARTHWORK   REMOVED   BY   CARTS. 

1.  Loosening  the  Earth  ready  for  the  Shovellers.  — 

A  two-horse  plough,  with  two  men  to  manage  it,  will  loosen 
about  250  yards  per  day  of  strong  heavy  soil,  about  500  yards 
of  common  loam,  or  about  1000  yards  of  light  sandy  soil  ; 
thus  the  cost  of  loosening  these  materials  per  cubic  yard  will 
respectively  be  about  1.5  cents,  0.8  cent,  and  0.4  cent  —  i.e., 
assuming  the  total  cost  of  the  plough  and  men  and  horses  con- 
nected with  it  to  be  about  $3.87  per  day.  When  a  four-horse 
plough  is  needed,  as  in  dealing  with  stiff  clays  or  cemented 
gravel,  the  cost  runs  up  to  about  2.5  cents  per  cubic  yard. 

Loosening  by  picks  costs  about  three  times  as  much  as  by 
ploughs,  where  the  latter  can  work  to  advantage.  The  amount 
which  a  man  can  loosen  with  a  pick  in  a  day  varies  from 
about  14  to  60  yards,  according  to  the  material. 

2.  Shovelling  the   loosened   earth    into    carts.  —  The 
shovellers  are  usually  actually  at  work  from  5  to  7  hours  out 
of  the  day.     If  we  assume  that  each  cart  carries,  as  a  working 
load,  £  cu.  yd.,  a  shoveller  can  load  it  in  from  5  to  7  minutes, 
according  to  the  nature  of  the  material  ;  and  suppose  he  is 
actually  shovelling  for  6  hours  out  of  the  day,  then  in  the 
course  of  the  10  hours    he  handles  about  24  yards  of  light 
sandy  soil,  20  yards  of  loam,  and  17  of  heavy  soil  at  the  cost 
of  4.2  cents,  5  cents,  and  5.8  cents,  respectively. 

3.  Hauling  away  the  earth,  dumping  and  returning. 
—  The  average  speed  of  horses  when  hauling  is  about  200  feet 
per  minute,  so  that  every  100  feet  of  lead  occupies  about  one 
minute  ;  dumping  and    turning    occupies  about  another  4 
minutes;   so  that  the  number  of  trips  per  cart  per  day  equals 


where  M  =  number  of  minutes  in  the  working  day  (here  600) 
and  L  =  length  of  the  lead  in  terms  of  100  feet.  Then  %N 
equals  the  number  of  cubic  yards  moved  by  each  cart  per  day  ; 
and  %N,  divided  into  the  total  expense  of  the  cart  per  day, 
gives  the  cost  of  hauling  per  cubic  yard.  Assuming  that  one 
driver  attends  to  four  carts  (doing  nothing  else),  the  total  cost 
per  cart  may  be  set  at  $1  .25  per  day. 


134 


KAILROAD   CONSTRUCTION. 


4.  Spreading  on  the  embankment.— The  cost  of  this 
varies  considerably,  but  may  be  said  to  average  about  1|  cents 
per  cu.  yd.  When  the  earth  is  dumped  over  the  end  of  the 
embankment,  or  is  "wasted,"  £  cent  per  cu.  yd.  should  be 
allowed  for  keeping  the  dumping-places  clear. 

Keeping  the  hauling  road  in  good  order. — This  is  an  item 
highly  expensive  if  neglected,  but  if  well  looked  after,  y1^ 
cent  per  cu.  yd.  per  100  feet  of  lead  is  usually  sufficient  to 
cover  it. 

Wear  and  tear  of  tools. — "Experience  shows  that  £  of  a 
cent  per  cubic  yard  will  cover  this  item."  This  also  includes 
the  interest  on  the  cost  of  the  tools. 

Besides  the  above,  1|  cents  per  cubic  yard  should  be  added 
to  cover  the  cost  of  superintendence  and  water-carriers,  and 
about  £  cent  for  extra  trouble  in  ditching  and  trimming  up. 

As  regards  the  profit  to  the  contractor,  it  may  be  set  down 
as  from  about  6  to  15  per  cent,  according  to  the  magnitude 
of  the  work  and  the  risks  incurred  ;  out  of  this  he  usually 
has  to  pay  the  clerks,  store-keepers,  cost  of  shanties,  etc.,  but 
these  as  a  rule  cover  their  own  expenses. 

The  following  table  gives  the  cost,  exclusive  of  profit  to  the 


Length  of 
Lead  in 
feet. 

Cu.  yds. 
hauled 
per  day  per 
cart. 

TOTAL  COST,  PLOUGHED  AND  SPREAD,  IN  CENTS. 

Light 
sandy  soil. 

Common 
loam. 

Qf™          Stiff  clay  or 

"ATon.  <~ 

50 

44.4 

10.4 

12.2 

13.7 

14.7 

100 

40.0 

10.8 

12.5 

14.0 

15.0 

200 

33.3 

11.5 

13.2 

14.8 

15.8 

300 

28.6 

12.2 

14.0 

15.5 

16.5 

400 

25.0 

12.5 

14.7 

16.2 

17.2 

600 

20.0 

14.4 

16.1 

17.7 

18.7 

800 

16.7 

15.8 

17.6 

19.1 

20.1 

1000 

14.3 

17.3 

19.0 

20.6 

21.6 

1200 

12.5 

18.8 

20.5 

22.0 

23.0 

1400 

11.1 

20.2 

21.9 

23.4 

24.4 

1600 

10.0 

21.7 

23.4 

24.9 

259 

1800 

9.1 

23.1 

24.8 

26.3 

27.3 

2000 

8.3 

246 

26.3 

27.8 

28.8 

2500 

6.9 

28.2 

29.9 

31.4 

32.4 

3000 

5.9 

31.8 

33.5 

35.0 

36.0 

4000 

4.5 

39.0 

40.8 

42.3 

43.3 

5000 

3.7 

46.4 

48.1 

49.6 

50.6 

RAILROAD   CONSTRUCTION. 


135 


contractor,  of  earth  when  ploughed  and  spread  in  the  embank. 
ment.  When  loosened  with  picks,  from  1.3  to  4.5  cents  per 
cu.  yd.  should  be  added  to  the  values  given,  according  as  to 
whether  the  material  is  of  a  light  sandy  nature  or  a  stiff  clay. 
If  merely  dumped  over  the  embankment,  then  the  values 
given  may  be  reduced  by  about  1  cent  per  cubic  yard. 

B.    THE    COST   OF   ROCK   REMOVED   BY  CARTS. 

The  total  cost  of  loosening  hard  rock  —  with  wages  at  $1.00 
per  day  —  is  usually  covered  by  45  cents  per  yard  in  place  ;  in 
dealing  with  soft  shales  which  can  be  loosened  by  pick,  being 
sometimes  as  low  as  20  cents,  while  in  shallow  cuttings  of 
tough  rock,  in  which  the  strata  lie  unfavorably,  $1.00  may  be 
insufficient. 

A  good  churn-driller  will  drill  from  8  to  12  feet  of  2-inch 
holes,  about  2|  feet  deep,  per  day,  at  a  cost  of  about  12  to  18 
cents  per  foot. 

A  cart  suitable  for  £  cu.  yd.  of  earth  as  a  working  load 
will  take  about  \  cu.  yd.  of  rock.  Rock  takes  longer  to 
shovel  into  the  carts  than  earth,  so  that  we  may  say  the  equa- 
tion given  above  for  earth  becomes  in  the  case  of  rock 


and  the  number  of  yards  hauled  per  day  is  given  by  ^N. 
Loading  costs  about  8  cents  per  cu.  yd.,  and  the  repair  of  the 
hauling-road  about  ^  cent  per  cu.  yd.  per  100  feet  of  lead. 
Thus  we  have,  exclusive  of  the  profit  to  the  contractor  — 


Length  of  Lead 
in  feet. 

No.  of  cu.  yds. 
per  cart  per 
day. 

Cost  per  cu.  yd. 
for  hauling  and 
emptying. 

Total  cost  per 
cu.  yd. 

50 

18.5 

6.8 

60.0 

100 

17.1 

73 

60.5 

200 

15.0 

8.3 

61.7 

300 

13.3 

9.4 

63.0 

500 

10.9 

11.5 

65.5 

700 

9.2 

13.6 

68.0 

1000 

7.5 

16.7 

71.7 

1500 

5.7 

21.9 

77.9 

2000 

4.6 

27.1 

84.1 

2500 

3.9 

32.3 

90.3 

3000 

3.3 

37.5 

96.5 

4000 

2.6 

47.9 

108.9 

136  RAILROAD   CONSTBUCTIOtf. 

"  Loose  Rock"  usually  costs  about  30  cents  per  yard  less 
than  the  above  cost  for  hard  rock. 

125.  Both  rock  and  earth  can  generally  be  moved  at  about 
the  same  cost  by  wheelbarrows  as  by  carts  when  the  lead  is 
equal  to  about  200  feet  ;  for  shorter  hauls  the  wheelbarrows 
have  the  advantage,  but  for  longer,  the  carts. 

As  regards  the  cost  of  removal  by  scrapers  or  any  other 
form  of  vehicle,  it  may  be  approximated  to  in  the  same  man- 
ner as  the  removal  by  carts  in  Sec.  124.  A  scraper  generally 
moves  from  30  to  60  cubic  yards  per  day  with  a  short  haul.  A 
medium-size  steam-shovel,  if  kept  tolerably  busy,  should,  un- 
der ordinary  conditions,  load  the  cars  at  a  cost  of  from  2  to  3 
cents  per  cu.  yd.'  Grading-machines,  8 or  12  horse,  in  light 
soil  and  with  low  fills,  can  generally  turn  over  from  500  to 
1000  cu.  yds.  per  day. 

126.  Estimating  Overhaul. — It  is  common  to  allow  an 

extra  price, usually  from  1 
to  2  cents  for  every  cubic 
yard  of  material,  either 
earth  or  rock,  for  each 
100  feet  that  it  is  hauled 
beyond  what  is  termed 
FlG-  67-  the  limit  of  free  haul, 

represented  by  I  in  Fig.  67. 

Let  us  suppose  that  the  material  in  the  cut  AC  is  just  suffi- 
cient to  make  the  fill  CB,  then  the  material  on  which  overhaul 
must  be  charged  is  that  lying  between  A  and  D  (or  B  and  E), 
and  the  distance  which  that  material  is  hauled  is  represented 
by  L,  the  distance  between  the  centres  of  gravity  of  the  two 
solids  AD  and  EB;  consequently  the  length  of  overhaul 
=  L  —  I,  and  if  8  represents  the  contents  of  AD  (or  EB),  then 
the  amount  of  overhaul  —  8(L  —  I). 

Thus,  for  example,  if  L  =  1000  ft.,  I  =  600  ft. ,  and  S  =  4000 
cu.  yds.,  the  cost  of  overhaul  at  1  cent  per  cu.  yd.  per  100  ft. 
will  be  |160. 

But  though  the  distance  I  is  always  given,  in  order  to  locate 
it  on  the  profile  we  must  find  the  points  D  and  E,  such  that 
the  material  in  DC  =  the  material  in  EG.  This  may  usually 
be  done  by  inspection  of  the  profile  ;  and  in  the  same  way 
the  points  A  and  B  may  be  fixed.  In  cases  where  the  centre- 
heights  are  not  fair  indications  of  volume,  these  points  may 


RAILROAD  CONSTRUCTION.  137 

be  quickly  found  to  within  a  few  feet,  by  means  of  the  cross- 
section  note-book.  The  positions  of  the  centres  of  gravity  of 
the  two  solids  AD  and  EB  may  also  usually  be  fixed  by  inspec- 
tion. On  this  subject  the  Engineering  News  says:  "  As  quick  a 
way  as  any  is  to  plot  the  volumes  of  each  solid  as  ordinates,  as  one 
would  plot  a  profile,  on  stiff  card-board,  cut  out  the  area  thus 
drawn,  and  balance  it  on  a  knife-edge ;  but  a  way  which 
we  can  recommend  as  much  the  best  and  fairest  of  any,  in 
competent  hands,  is  to  guess  at  it,  throwing  the  benefit  of  a 
doubt  for  or  against  the  contractor  according  to  the  character 
of  the  haul,  and  to  some  extent  of  the  material  excavated. 
The  actual  haul  cannot  fairly  be  taken  at  times  as  the  crow 
flies,  nor  is  it  exactly  fair  that  haul  over  good  solid  gravel 
should  have  the  same  allowance  as  haul  from  a  shallow  cut 
through  muck.  As  a  contract  is  a  contract,  and  must  be  gen- 
eral, no  considerable  deviations  on  account  of  such  contin- 
gencies as  these  are  admissible,  but  no  considerable  ones  are 
necessar}',  the  limits  of  error  in  guessing  at  the  '  centre  of 
mass  '  being  very  small,  and  having  reference  to  a  small  item 
of  price,  whereas  the  limits  of  error  in  one  unavoidable  kind 
of  guessing  which  is  usually  going  on  at  the  same  time,  that 
of  classification,  are  very  large,  and  have  reference  to  a  very 
large  item.  This  consideration  alone  ought  to  show  the  folly 
of  any  great  hair-splitting  in  mathematical  computations  of 
the  precise  overhaul  ;  but  there  is  a  certain  class  of  minds 
who  are  never  happy  unless  they  can  find -some  hair  to  split, 
and  who  will  split  it  with  just  as  much  care  although  there 
may  be  a  log  of  wood  alongside  which  they  can't  split,  to 
which  the  right  half  of  the  hair  is  to  be  added." 

THE  CALCULATION  OF  EARTHWORK. 

127.  The  three  solids  with  which  engineers  have  mainly  to 
deal  in  the  calculation  of  earthwork  are  the  pyramid,  the 
wedge,  and  the  "  prism oid  ;"  for  though,  owing  to  the  irregu- 
larities of  surface,  these  figures,  mathematically  speaking,  are 
never  actually  met  with  in  practice  where  the  surface  of  the 
ground  forms  one  or  more  sides  of  the  figure,  yet  the  contents 
as  given  by  them  are  sufficiently  accurate  under  ordinary  cir- 
cumstances, when  the  work  has  been  properly  cross-sectioned. 
But  before  dealing  with  the  calculation  of  the  contents  of 


133 


RAILROAD  CONSTRUCTION. 


these  solids,  it  will  be  well  to  consider  the  methods  of  obtain- 
ing the  areas  of  the  cross-sections  themselves,  on  which 
the  computations  are  based. 

1.  When  the  cross-section  is  of  triangular  form,  as  in  Fig.  69, 
its  area  of  course  —  taking  for  instance  the  triangle  ABO  — 
equals  AB  X  1  the  perpendicular  distance  from  C  to  AB,  or 
AB  produced. 

2.  When  the  cross-section  is  an  ordinary  3-level  one,  as  in 
Figs.  71  and  72,  then  if  B  =  width  of  road-bed  and  //,  h,  7i',  I, 
and  I  '  are  as  shown  in  Fig.  55, 


which  is  the  formula  most  generally  in  use. 
3.  If  the  surface  is  horizontal,  then  this  becomes 


Area  =  1 
4.  Or,  if  regularly  inclined, 


where  h  is  the  greater  side-height,  and  I  its  corresponding 
distance  out  from  the  centre,  h'  being  the  smaller  side-height. 

5.  But  it  frequently  happens  that  we  have  such  a  section  as 
that  shown  in  Fig.  68.     Such  an  area  may  be  best  calculated 


iL._\ 


FIG.  68. 

by  first  finding  the  contents  of  the  figure  IDHL,  and  then 
deducting  from  it  the  areas  DIA  and  HLB;  thus  the  area  of 
this  cross  section  equals 


TD-\-EJ 
2 


EJ+FC 


2 


GK+HL 


FC+GK 

2 
fjrT       ID.IA      BL.HL 

(AL)  — 


2  2  2 

The  above  forms  of  cross  sections  are  really  all  that  are  re- 
quired in  practice,  1,  2,  and  5  being  those  most  generally  in 


EA1LEOAD   CONSTRUCTION.  139 

use.  Neither  of  these  forms  requires  plotting,  but  it  is  usually 
advisable  to  plot  cross-sections  of  large  area  which  are  very 
irregular  even  though  calculated  as  above,  for  by  so  doing 
mistakes  are  much  more  readily  apparent.  Where  the  work 
consists  largely  of  irregular  cross  sections,  a  good  and  rapid 
method  of  obtaining  the  areas  is  to  plot  the  cross-sections  and 
use  a  planimeter.  The  error  in  ordinary  cross-sections, 
plotted  on  cross  section  paper  to  a  scale  of  10  feet  to  an  inch, 
should  never — where  the  planimeter  is  carefully  adjusted  so  as 
to  allow  for  the  shrinkage  of  the  paper,  etc. — exceed  1  p.  c. ; 
and  considering  that  these  errors  to  a  large  extent  cancel  each 
other  and  are  free  from  errors  of  calculation,  which  are 
usually  much  more  probable  than  errors  in  reading  the  planim- 
eter scale,  the  result  in  the  long  run  is  at  least  equally  likely 
to  be  as  near  the  truth  as  that  obtained  by  the  more  laborious 
process  of  calculation. 

128.  The  areas  of  the  cross-sections  having  been  obtained, 
the  calculation  of  the  contents  of  the  solids  which  they  bound 
is  the  next  point  to  deal  with,  and  we  will  consider  them  in  the 
order  given  above. 

A.  The  Pyramid.— The  usual  cases  in  which  pyramids 
occur  are  those  shown  in  Fig.  69,  which  need  no  explanation. 


Fro.  69. 

The  contents  of  such  a  pyramid  as  ABCD  are  found  by  the 
formula 

S=ABCX~, 

and  this  rule  applies  to  any  form  of  base. 

B.  The  Wedge.— The  various  forms  of  wedge  which  pre- 
sent themselves  in  calculating  the  contents  of  earthwork,  of 
which  that  represented  in  Fig.  70  is  the  usual  type,  can  only 
be  estimated  correctly  by  the  application  of  the  Prismoidal 


140  RAILROAD   CONSTRUCTION. 

Formula.     But  since  at  the  points  where  the  wedge  form  of 
solid  occurs  the  cut  or  .fill  is  always  small,  the  error  involved 


by  using  the  formula  for  the  rectangular  wedge  is  immaterial; 
thus  we  may  say  that  the  contents 

A  fl- 
8  =  area  ABODE  X  ~:- 

C.  The  Prismoid.—  Though  the  term  "  prismoid  "  strictly 
applies  only  to  such  solids  as  are  contained  by  6  plane  surfaces, 
the  two  end  -faces  being  parallel,  and  two  of  the  other  faces 
being  not,  parallel,  the  extended  application  of  the  "  pris- 
moidal  formula"  has  corrupted  its  true  meaning,  so  that  it  is 
now  applied  very  generally  in  Railroad  work  to  all  solids  hav- 
ing two  parallel  faces,  whether  plane  or  curved,  upon  which, 
and  through  every  point  of  which,  a  straight  line  may  be 
drawn  from  one  of  the  parallel  faces  to  the  other. 

The  contents  of  such  a  solid  according  to  the  PRISMOID  AL 
FORMULA  equal 


where  L  =  the  length  of  the  solid, 

A  and  a  =  the  areas  of  its  two  parallel  faces, 

and  M=  the  cross-section  parallel  to  A  and  a,  and  half-way 
between  them. 

This  formula  at  first  looks  simple  enough,  but  the  calculation 
of  M  is  the  difficulty. 

129.  To  explain  the  application  of  this  formula,  suppose  we 
have  two  end-areas  A  and  a  as  in  Fig.  71. 

Now  in  order  to  obtain  the  mid  section,  we  must  know  the 
points  in  A  and  a  from  which  the  straight  lines  joining  them 
start,  and  at  which  they  end;  thus  in  Fig.  71,  if  the  cross- 


RAILROAD   CONSTRUCTION". 


141 


section  notes  simply  give  the  elevations  for  the  3-level  sections 

A  and  a,  we  assume  that  the  upper  surface  between  them  is 

B 


Fio.  71. 

composed  of  two  warped  surfaces,  BCcb  and  GDdc,  which  is 
what  follows  from  supposing  that  the  centre  and  side  heights 
of  M  are  the  averages  of  the  corresponding  heights  of  A  and  a. 
So  that  if  the  surface  were  actually  as  shown  in  Fig.  72, 


FIG.  72. 

we  should  obtain  entirely  erroneous  results  by  taking  the 
value  of  M  given  by  Fig,  71.  Thus  when  the  surface  is 
such  that  points  in  A  and  a,  other  than  those  directly  cor- 
responding, are  to  be  considered  as  being  joined  by  straight 


142  RAILROAD   CONSTRUCTION-. 

lines,  it  becomes  necessary  to  indicate  in  the  notes  between 
what  points  in  A  and  a  the  straight  lines  are  assumed  to  be 
drawn;  and  then  the  surface,  instead  of  being  made  up  of  two 
or  more  warped  surfaces,  will  be  composed  entirely  of  a  series 
of  plane  surfaces  as  in  Fig.  72.  This  is  best  done,  where  re- 
quired, by  drawing,  in  the  cross-section  note-book,  lines  con- 
necting the  notes  of  the  points  to  be  joined.  This  would  also 
have  to  be  done  between  two  cross  sections  A  and  a  which  did 
not  happen  to  have  the  same  number  of  points  taken  in  each. 
At  times  cases  occur  in  which  it  is  advisable  to  fill  in  slope- 
lines  in  this  way,  but  they  are  very  few  and  very  far  between; 
for  the  labor  involved  in  the  calculation  of  M  in  such  cases 
would  usually  have  been  very  much  better  expended  in  actually 
taking  a  cross-section  between  A  and  a.  Therefore,  as  a  rule, 
where  the  prismoidal  formula  is  to  be  used  in  the  calculation 
of  the  contents,  it  is  very  much  better  to  cross-section  a  little 
more  closely,  where  necessary,  and  to  omit  the  filling-in  of 
the  slope-lines,  than  to  take  cross-sections  a  little  farther  apart 
and  fill  in  the  slope-lines  by  inspection. 

The  value  of  the  prismoidal  formula,  as  applied  in  the  case 
of  Fig.  71,  is  not  so  much  to  rectify  irregularities  in  surface  as 
to  make  suitable  allowance  for  the  difference  in  the  heights  of 
A  and  a,  which  the  method  of  average  end-areas  does  not  do. 
In  practice,  however,  where  the  work  is  properly  cross-sectioned, 
the  application  of  the  prismoidal  formula  is  a  mathematical 
refinement  which  is  entirely  unnecessary,  for  the  method  of 
average  end-areas — that  usually  employed — then  gives  results 
sufficiently  satisfactory,  both  to  the  Railway  Company  and  the 
Contractor. 

It  is  an  interesting  fact  in  connection  with  Figs.  71  and  72, 
that  if  the  contents  be  calculated  for  each  possible  arrangement 
of  slope-lines,  the  mean  of  the  results  so  obtained  will  be 
equal  to  the  result  as  derived  by  merely  the  joining  of  corre- 
sponding points,  as  in  Fig.  71. 

The  calculation  of  the  mid-area  is  merely  a  matter  of  simple 
proportion.  In  dealing  with  such  a  case  as  Fig.  72,  by  plotting 
A  and  a  on  a  sheet  of  cross-section  paper,  the  drawing  of  the 
mid-sections  may  be  done  by  simply  drawing  parallel  lines  ;  so 
that  this  should  be  done  as  a  check  to  the  calculations  and  also 
as  a  means  of  facilitating  them. 


RAILROAD   CONSTRUCTION.  143 

130.  The  method  used  nowadays  almost  entirely  for  the 
calculation  of  grading,  is  that  of  Average  End-areas,  which 
assumes  that 

,      A  +  a 


Now  this  method,  which  is  the  simplest  of  any  to  work, 
unfortunately  has  a  considerable  tendency  to  excess;  the  re- 
sults obtained  by  it  are,  however,  the  same  as  those  given  by 
the  prismcidal  formula  —  applied  as  in  Fig.  71,  —  therefore 
presumably  correct,  under  the  following  circumstances  : 

1.  Whenever  the   centre-heights  of  A  and  a  are  the  same, 
whatever  the  difference  in  side  heights  may  be. 

2.  Whenever  the  entire  widths  between  the  slope  stakes  at 
A  and  a  are  the  same,  whatever  the  difference   in   centre- 
heights  may  be. 

When,  however,  the  smaller  centre-height  is  at  the  same 
end  of  the  solid  as  the  greater  width  between  the  slope-stakes, 
the  volume  as  given  by  average  end-areas  will  be  actually  de- 
ficient. 

But  since  these  cases  are  the  exceptions,  the  results  as  given 
by  this  method  are  in  the  long  run  considerably  too  high, 
unless  care  is  taken  in  cross-sectioning  to  limit  the  excess.  To 
correct  for  this  tendency  a  Prismoidal  Correction  maybe 
used,  found  by  deducting  the  prismoidal  formula  from  the 
formula  for  average  end-areas  ;  and  this  correction,  when  the 
surface  of  each  end-section  is  horizontal,  equals  in  cubic  yards 


where  ZTand  H'  are  the  end  centre-heights  in  feet,  s  the  slope- 
rates,  and  L  the  lengths  of  the  solid  in  feet. 

Taking  s  =  U  and  L  —  100,  we  obtain  the  following  values 
for  C,  which  serve  in  making  up  preliminary  estimates  to 
show  the  errors  involved  by  a  rough  system  of  cross-sectioning 
when  the  contents  are  calculated  by  average  end-areas. 


144 


RAILROAD    CONSTRUCTION. 


TABLE  OF  PRISMOIDAL  CORRECTION  FOR  100  FEET 

IN  CU.  YDS.  FOR  HORIZONTAL  SURFACES 

WHERE  s  =  U. 


//  -  //' 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

834 

779 

0 
10 
20 

0 
93 
370 

1 
llii 

408 

4 
133 

448 

8 
156 
490 

15 
181 
533 

23 

208 
578 

33 
237 
626 

45 

268 
675 

59 

300 

720 

This  value  of  C  is  altogether  independent  of  the  width  of  the 
road-bed  ;  so  that,  for  example,  suppose  on  ground  sloping  in 
the  direction  of  the  length  of  the  solid  we  have,  between  two 
sections  100  feet  apart,  a  dill'ereuce  in  centre-heights  of  23  feel, 
if  s  =  \\  and  there  is  no  slo{>e  transversely,  the  contents  as 
given  by  average  end-areas  will  be  490  cubic  yards  too  much, 
even  with  a  14-foot  road-bed  ;  or.  if  the  fill  at  one  end  is  2  feet 
and  at  the  other  end  25  feet,  the  prismoidal  formula  gives 
1937  cubic  yards  as  the  volume,  while  the  method  of  average 
end-areas  gives  2447  cubic  yards,  or  25  p.  c.  too  much. 

But  the  above  values  of  the  prismoidal  correction  only  apply 
when  the  surfaces  of  the  sections  are  horizontal.  If,  however, 
in  dealing  with  3-level  sections  we  call  Wand  W  the  entire 
width  between  the  slope-stakes  at  each  end,  then  the  prismoi- 
dal correction  equals,  in  cubic  yards, 


which  is  independent  of  the  side-slopes  and  width  of  the  road- 
bed. So  that,  having  calculated  the  contents  according  to  the 
formula  for  average  end-areas,  we  have  simply  to  find  for 
each  cross-section  the  value  of  (H  '—  H'}  and  (W  -  W'},  and 
take  out  from  the  following  table,  which  gives  the  values  of  C, 
the  amount  in  cubic  yards  which  is  to  be  added  to  the  contents 
already  obtained  in  order  to  obtain  the  result  which  would  be 
given  by  the  prismoidal  formula.  Should,  however,  the 
smaller  centre-height  be  at  the  same  end  of  the  solid  as  the 
greater  width  between  the  slope  slakes,  then  C  must  be  sub- 
tracted. 


BAILROAD   CONSTRUCTION. 


145 


TABLE  OF  THE  VALUES  OF  G,  WHEN  L  =  100  FEET, 


W-W 
in  feet. 

H—  H'  in  feet. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

1 

.3 

.6 

.9 

1.2 

1.5 

1.8 

2.1 

2.4 

2.7 

3.1 

2 

.6 

1.2 

1.8 

2.4 

3.0 

3.6 

4.3 

4.9 

5.5 

6.2 

3 

.9 

1.8 

2  7 

3.6 

4.6 

5.5 

6.5 

7.4 

8.3 

9.3 

4 

1.2 

2.4 

3^6 

4.9 

6.2 

7.4 

8.6 

9.8 

11.1 

12.3 

5 

1.5 

3.1 

4.6 

6.2 

7.7 

9.2 

10.8 

12.3 

13.8 

15.4 

6 

1.8 

3.6 

5.5 

7.4 

9.2 

11.1 

12.9 

14.8 

16.6 

18.5 

7 

2.1 

4.3 

65 

8.6 

10.8 

12.9 

15.1 

17.3 

19.4 

21.5 

8 

2.4 

4.9 

7.4 

9.8 

12.3 

14.8 

17.3 

19.7 

22.2 

24.6 

9 

2.7 

5.5 

8.3 

11.1 

13.8 

16.6 

19.4 

22.2 

25.0 

27.7 

10 

3.1 

6.2 

9.3 

12.3 

15  4 

18.5 

21.5 

24.6 

27.8 

30.8 

11 

3.4 

6.8 

10.2 

13.6 

17.0 

20.3 

23.7 

27.1 

30.6 

33.9 

12 

3.7 

7.4 

11.1 

14.8 

18.5 

22.2 

25  8 

29.5 

33.3 

37.0 

1| 

4.0 

8.0 

12.0 

16.0 

20.0 

24.0 

28.0 

32.0 

36.0 

40.1 

14 

4.3 

8.6 

12.9 

17.3 

21.5 

25.8 

30.1 

34.5 

38.8 

43.2 

15 

4.6 

9.2 

13.8 

18.5 

23.1 

27.7 

32.3 

37.0 

41.6 

46.3 

16 

4.9 

9.8 

14.8 

19.7 

24.6 

29.5 

34.5 

39.4 

44.3 

49.3 

17 

5.2 

10.4 

15.7 

20.9 

26.2 

31.4 

36  6 

41.9 

47.1 

52.4 

18 

5.5 

11.1 

16.7 

22.2 

27.8 

33.3 

38.8 

44.4 

49.9 

55.5 

?9 

5.8 

11.7 

17.6 

23.4 

ao.s 

35.1 

41.0 

46.9 

52.7 

58.6 

20 

6.2 

12.3 

18.5 

24.6 

30.8 

37.0 

43.2 

49  4 

55.6 

61.8 

21 

6.5 

12.9 

19.4 

25.8 

32.3 

38.8 

45.3 

51.8 

58.3 

64.8 

22 

6.8 

13.5 

20.3 

27.1 

33.9 

40.6 

47.4 

54.3 

61.1 

67.9 

23 

7.1 

14.2 

21.3 

28.4 

35.4 

42.5 

49  6 

56.8 

63.9 

71.0 

24 

7.4 

14.8 

22.2 

29.6 

37.0 

44.4 

51.8 

59.2 

66.7 

74.1 

25 

7.7 

15.4 

23.1 

30.8 

38.5 

46.2 

54.0 

61.7 

69.4 

77.1 

26 

8.0 

16.0 

24.0 

32.0 

40  0 

48.1 

56.1 

64.1 

72.1 

80.2 

27 

8.3 

16.6 

24.9 

33.2 

41.5 

49.9 

58.3 

66.6 

74.9 

83.3 

28 

8.6   17.2 

25.8 

34.5 

43.1 

51.8 

60.5  69.1 

77.7 

86.4 

29 

8.9  17.8 

26.8 

35.7 

44.7 

53.7 

62.7  71.6  80.5 

89.5 

30  j  9.3 

18.5 

27.7 

37.0 

46.3 

55.6 

64.9 

74.1 

83.3 

92.6 

There  is  no  need  to  apply  these  corrections  at  the  time  when 
the  quantities  are  worked  out  by  average  end-areas,  as  generally 
the  engineer  is  then  too  much  occupied  in  obtaining  rougJi  es- 
timates of  the  work  ;  but  they  can  subsequently  be  applied, 
with  very  little  trouble,  to  such  solids  as  in  his  opinion  need 
correcting. 

The  application  of  this  method  undoubtedly  reduces  the 
final  estimate  of  the  grading  very  considerably,  rarely  by  less 
than  1  p.  c.,  and  in  some  cases,  where  the  cross-sectioning  has 
been  carelessly  done,  by  as  much  as  4  or  5  p.  c.  But  it  must 
be  remembered  that  in  this  way  the  true  volume  is  obtained 
more  nearly  than  by  any  other  of  the  approximate  processes, 
and  that  the  results  are  slightly  higher  than  those  obtained  by 
the  use  of  such  tables  as  "  Trautwine,"  "  Rice,"  etc.,  founded 
on  the  principle  of  Equivalent  Level  Sections.  Without  the 


146  RAILROAD   CONSTRUCTION. 

application  of  the  prismoidal  correction  the  contractor  is  en- 
tirely at  the  mercy  of  the  engineer  who  does  the  cross-section- 
ing (if  the  method  of  average  end-areas  is  used),  who  has  it, 
often  unconsciously,  in  his  power  to  make  a  difference  in  the 
final  estimate  of  3  or  4  per  cent,  by  not  paying  attention  to 
the  differences  in  centre-heights  and  widths  of  the  cross-sec- 
tions he  is  taking.  And  though  the  errors  in  any  given  piece 
of  work  are  in  favor  of  the  contractor,  still  the  uncertainty 
to  which  they  give  rise,  in  the  long  run  do  him  considerably 
more  harm  than  good.  If  a  correction  is  not  used,  some 
limiting  value  for  (H—H)  X  (W -  W)  should  be  estab- 
lished. 

Some  standard  system  of  measuring  grading  is  much  wanted. 
As  it  is  now,  a  contractor  on  one  piece  of  work  gets  the  benefit, 
possibly  of  3  p.  c.  due  to  the  use  of  average  end -areas,  uii- 
corrected;  while  on  the  next  contract  he  takes  very  likely  he 
has  the  quantities  actually  cut  down,  owing  to  the  use  of  tables 
of  equivalent  level  sections.  It  is  true  that  if  the  work  is 
properly  cross-sectioned  the  excess  as  given  by  the  method  of 
average  end-areas  should  not  exceed  1  or  2  p.  c.,  but  in  the 
ordinary  way  in  which  cross  sectioning  is  done,  a  considerable 
amount  of  trouble  is  taken  in  order  to  correct  for  small  sur- 
face irregularities,  while  the  great  errors  which  are  involved 
by  the  difference  in  centre-heights  are  barely  considered  so 
long  as  the  slopes  between  the  sections  are  tolerably  uniform. 

When  the  cross-sections  are  irregular,  the  prismoidal  correc- 
tion can  usually  be  applied  with  sufficient  accuracy  by  treating 
them  as  3-level  sections,  and  thus  applying  the  value  of  C  as 
given  above. 

131.  The  Method  of  Equivalent  Level  Sections  is  an 
incorrect  means  of  applying  the  prismoidal  formula  by  reduc- 
ing the  end-sections  to  sections  equivalent  in  area  but  with 
their  surfaces  horizontal,  and  then  taking  as  the  area  of  the 
mid-section  that  which  is  given  by  the  mean  of  the  corrected 
centre-heights.  But  unfortunately  the  results  so  obtained  are 
only  correct — 

1.  When  the  two  end -areas  are  "  similar"— i.e.,  the  corre- 
sponding surface-slopes  from  the  centre  to  the  slope-stakes  are 
the  same  at  both  ends,  provided  the  road-bed  is  not  intersected 
between  them; 

2.  When  the  surface  is  regularly  warped  from  one  end  to 


RAILROAD    CONSTRUCTION.  147 

the  other,  provided  that  no  two  of  the  straight  lines  connecting 
corresponding  points,  such  as  A,  a,  etc.,  in  Fig.  71  are  inclined 
to  grade  in  opposite  direction  (as  they  are  in  Fig.  71). 

In  cases  where  these  conditions  do  not  hold,  then,  assuming 
that  the  true  result  is  given  by  the  prismoidal  formula  if  merely 
the  corresponding  points  A,  a,  etc.,  are  joined  by  straight  lines, 
the  method  of  equivalent  level  sections  gives  results  too  small. 
But  if  the  surface  is  intersected  by  undulations,  running 
obliquely,  necessitating  the  use  of  "slope-lines"  as  in  Fig.  72, 
then  the  results  may  either  be  too  small  or  too  great,  according 
to  circumstances.  But  since  this  latter  method  of  applying 
the  prismoidal  formula  is  the  exception,  and  the  results  as 
obtained  by  applying  it  in  the  manner  shown  in  Fig.  71  more 
generally  correct,  the  general  tendency  of  the  method  of 
equivalent  level  sections  is  to  deficiency,  but  not  by  an  amount 
usually  sufficient  to  warrant  the  use  of  a  correction.  The  real 
objection  to  this  method  is  the  labor  involved  in  applying 
it  when  dealing  with  cross-sections  in  the  slightest  degree 
"irregular,"  and  even  in  dealing  with  3-level  sections  the 
work  involved  is  greater  than  that  by  the  method  of  average 
end-areas,  corrected;  while  the  result  in  the  former  case  is  an 
approximation,  in  the  latter  it  is  presumably  correct. 

132.  The  method  of  centre-heights,  which  is  very  useful 
in  making  preliminary  estimates,  simply  assumes  that  the  con- 
tents between  any  two  cross-sections  are  given  according  to 
the  method  of  average  end-areas,  the  area  at  each  end  being 
taken  as  the  area  of  a  horizontal  section  with  a  height  equal 
to  the  actual  centre-height.     The  results  so  obtained  naturally 
err,  sometimes  in  excess  and  sometimes  in  deficiency — the 
tendency  in  the  former  direction  being,  however,  the  more 
common.     But  since  there  is  no  decided  tendency  to  cumula- 
tive error,  the  result  obtained  as  a  whole-  for  several  stations 
where  the  direction  of  the  surface  slope  is  varied,  agrees  toler- 
ably well  with  the  true  volume,  though  for  anyone  station  the 
error  may  be  very  considerable.    In  the  long  run  more  ac- 
curate results  are  usually  given  by  this  method  than  by  that 
of  average  end-areas.     (See  Sees.  69  and  70.) 

133.  By  the  use  of  Table  XIV  the  labor  of  applying  the 
method  of  Centre-heights  is  greatly  reduced. 

Table   XV  saves   considerable  labor  in  reducing  areas  to 
cubic  yards,  by  avoiding  the  necessity  of  multiplying  by  100 


148 


KAILROAD   CONSTRUCTION. 


and  dividing  by  27.  There  is  no  need  to  take  the  quantities  out 
closer  than  to  the  nearest  yard.  In  using  the  table  for  lengths 
other  than  100  feet  a  good  deal  of  trouble  may  be  saved  in 
the  way  of  multiplication  and  division  by  reducing  each  time 
the  simpler  of  the  two  values  with  which  the  table  is  entered; 
thus  if  we  have  an  average  area  of  634  square  feet  for  50  feet, 
the  amount  opposite  317  gives  the  quantity  required,  instead 
of  dividing  2348.2  by  2. 

134.  Correction  for  Curvature.— We  have  hitherto  as- 
sumed that  the  cross-sections  are  parallel  to  each  other — i.e., 
that  the  track  is  straight.  Suppose,  however,  that  in  Fig.  73, 
exaggerated  for  the  sake  of  clearness,  o  represents  the  centre 
of  a  certain  curve  whose  radius  —  R,  the  cross-section  ACaB 
representing  any  cross-section  on  the  curve. 

Now  it  is  clear  that  if  we  have  two  cross-sections  whose 
centres  are  100  feet  apart  (along  the  curve)  and  take  in  each  a 
point  b,  situated  outside  the  centre  by  a  distance  y,  the  distance 
between  these  two  selected  points,  measured  along  a  line 
parallel  to  the  centre-line,  is  to  100  feet  as  R  -j-  y  is  to  R,  arcs 


FIG.  73. 

subtended  by  equal  angles  at  the  centre  being  proportional  to 
their  radii.  But  instead  of  calculating  the  contents  for  the 
varying  distance,  it  is  simpler  to  assume  that  the  track  is 
straight,  and  to  correct  the  sections  themselves  so  as  to  allow  for 
it :  so  that,  instead  of  using  the  above  proportion,  we  may 
consider  that  the  area  of  a  section  at  any  distance  y  from  the 
centre  must  be  increased  or  decreased  in  the  proportion 


x'  = 


x(R  ± 


where  x'  represents  the  corrected  area  and  x  the  original  area; 
y  being  positive  if  falling,  as  in  Fig.  73,  on  the  outside  of  the 
curve,  and  negative  if  falling  inside.  So  that  if  at  any  point 
as  a  we  measure  the  ordinate  x  and  its  distance  from  the 


RAILROAD   CONSTRUCTION  149 

centre  y,  the  above  equation  gives  us  x' ,  the  corrected  length 
of  x,  which,  being  measured  upwards  from  the  point  b,  gives  us 
a',  the  new  position  of  a.  Similarly  by  finding  other  positions 
of  a',  the  curved  line  ACa'B  being  drawn  through  them,  gives 
the  equivalent  section  on  a  straight  track. 

In  curves  of  8°  and  upwards,  where  the  slope  is  compara- 
tively steep  in  one  direction,  this  correction  should  be  applied. 
It  is  best  to  assume  an  average  section  for  two  or  three  stations 
together,  and  to  divide  the  radius  by  10,  so  as  to  make  R  a 
distance  easily  scaled,  and  then  to  divide  the  correction  so  ob- 
tained by  10.  Thus,  if  the  section  is  taken  as  an  average  one 
for  300  feet  on  a  10°  curve,  we  plot  R  =  57  feet,  and  the  cor- 
rection so  obtained — which  is  of  course  equal  to  the  difference 
between  the  contents  given  by  the  actual  section  and  the 
equivalent  section— must  itself  be  divided  by  10;  or,  what  is 
the  same  thing,  be  considered  to  apply  only  to  a  length  of  30 
feet.  Two  or  three  ordiuates  are  usually  sufficient  to  locate 
with  sufficient  accuracy  the  surface  of  the  equivalent  section. 
Where  the  surface  is  level  there  will  of  course  be  no  correction 
necessary,  for  then  the  excess  on  one  side  of  the  centre-line 
balances  the  deficiency  on  the  other. 

This  method  is  equally  easy  to  apply  to  any  form  of  cross- 
section,  however  irregular  it  may  be. 

135.  The  contents  of  the  toe  of  a  dump  are  commonly 
calculated  according  to  the  formula  given  in  Sec.  128  for  a 
wedge,  but  the  result  so  obtained  is  always  considerably  too 
small;  neither  can  the  prismoidal  formula  be  directly  applied. 


FIG.  74. 


First,  let  us  assume  the  surf  ace  of  the  ground  to  be  level;  then 
the  simplest  way  to  obtain  correctly  the  contents  of  the  toe  is 
to  consider  each  corner  as  a  quarter  of  a  cone;  then  if  H equals 
the  height  of  the  fill  in  feet,  and  s  the  slope  ratio,  the  contents 


150  RAILROAD   CONSTRUCTION. 

of  the  two  corners  together  equal 

.523/W; 
so  that  the  entire  contents  of  the  toe  are  given  by  the  formula 

8  =  . 


B  being  the  width  of  the  road-bed  in  feet.    This  formula  is 
easily  worked  out  by  means  of  Table  VIII.     8  must  then  be 
divided  by  27  to  reduce  it  to  cubic  yards. 
If  «  =  1|,  then  the  above  equation  becomes 


But  when  the  ground  slopes  downward  in  the  direction  of  the 
toe,  as  is  the  more  common  case,  then  we  may  consider  the 
toe  to  be  divided  into  two  portions,  as  shown  in  Fig.  74;  the 
upper  one,  which  we  have  just  dealt  with,  having  a  vertical 
height  equal  H,  and  the  lower  one  with  a  vertical  height  =  h. 
Then,  omitting  for  a  moment  the  consideration  of  the  circular 
corners,  the  contents  of  the  upper  portion  are  to  the  contents 
of  the  lower  portion  as  His  to  h.  Now,  though  this  does  not 
quite  hold  good  when  taking  the  corners  into  account,  the 
error  involved  by  assuming  it  to  do  so  is  immaterial;  so  that 
we  may  say,  that  when  the  ground  slopes  forward  as  in  Fig. 
74,  the  total  contents  equal 


rh 

the  value  of  S  being  obtained  as  above. 

The  value  of  h  may  be  obtained  quite  well  enough  by  plot- 
ting //and  the  slopes  of  the  ground  and  the  dump. 

If  the  ground  slopes  transversely  as  well,  the  case  becomes 
decidedly  complicated,  and  the  engineer  must  then  assume 
such  values,  as  will  when  inserted  in  the  above  formulae,  give 
what  he  considers  fair  results. 

In  dealing  with  the  toe  of  a  dump  less  than  10  feet  in  height 
the  wedge  formula  is  sufficiently  accurate,  but  where  the  fill 


RAILROAD  CONSTRUCTION". 


151 


t^c^ot-^oot-^oc^ 

as  :o  -  »  55  i  -^  i  ?  -  .-f  i  -  o  co  £-  o  co  i~  o  co  o  r- 

O  CO  O  «0  CO  O  to  10  O  00  CC i  O  CO  50  5  CO  CO  0  CO  0  CO 


co*?555*-«oQOcD2scS'"*s5  toccot>cooo-coot--oco 
i^t-coOiooo^ac^p^coSoGop^coXopco 

THrHC^C7SO^J1'^lAO»O?Dt;'-OOOiO'r-t'??Tt*iCCDt--CO'*-<CO 

c*  iiiliiilsliiiiiissssssg 

TH  i-<  sxj  oj  co  co  TT  ^  in  in  co  t-  ou  oi  o  i->  e>}  co  •**<  in  co  oo  o 

°  00 

•  i-c^ciWNcoco'^'Ttininot-oooJOO^-'TJcointo 

•  i-i  T-I  «  si  M  TO  co  eo'  TI<  TJ<  in  co  co' t-  oo  oa  os  o  •-<  o»  so  o 

coocotoocooococoo^cdocBTOococoStooco 

•  i-<  i-<  i-i  w  ^  w  co'  co'  co  •*  -^  ic  CD'  co  i-  c»  oo  oi  o  o  o*  eo' 

*"          ^^•^^^^^c.Vco^^ir'in'cot-^ 

•OOO'OO^OOOOOOOOO': 

M 

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.-  W  CO  —  1C  O  »C  05  TT  GO  CO  <N  i-c 

TO'  Tf  rr  in  in  m'  co'  CD'  t>  co  gi 

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tt       ^-r-n-iT-n-i|MO??>o<eoco^«^'  m  m  ic  co  to  t-  oo 

'     i-  jb  in  woi  rt  e  oo  i-  co  in  o>  S  f-  55  e»  o  f-  i~o  "^  o  J:  o 

co  in  i-  os  r-  s>j  ic  to  x  S  01  to  c  co  i-  »-i  ic  GO  oj  co  o  i-  o 

*T-.'T-ii-<'^^i^iyi?»eo'go';crii-^Tfirix;«>mt^ 

i^  CC  T?  O?  O 

eo 

c\i  c<?  m  to  i-  oo  o  1-1  ^_  co_  m  i- 

co  rr  in  to  t-  60  os  o  r-  c^  ^e  oo  o  <?J  in  4-  os  "  co  i-  S 

—  o  co  c— 

o          1-1  ^Tc*  c^co  «T-< 


152  RAILROAD   CONSTRUCTION. 

amounts  to  about  20  feet  the  difference  in  the  results  by  the 
two  methods  is  very  considerable. 

136.  The  original  notes  of  the  cross-sections  should  be  copied 
on  the  left-hand  pages  of  another  note-book,  and  opposite 
them,  on  the  right-hand  pages,  the  sectional  areas,  contents, 
etc.,  should  be  entered  as  soon  as  worked  out.  A  "Record" 
should  also  be  kept,  into  which  each  separate  item  should  be 
entered  as  soon  as  completed, — not  in  detail,  but  simply  the 
total  amounts;  these  notes  then  form  the  groundwork  of  the 
final  estimate.  The  details  are  entered  separately  in  note- 
books apportioned  to  each  class  of  work. 

As  regards  taking  notes  for  the  monthly  estimates,  the 
simplest  way  is  to  walk  over  the  work  and  sketch  on  the  prog- 
ress profile  the  state  of  construction  at  the  time.  Another 
way,  possibly  more  convenient  in  light  work,  is  to  note  the 
percentage  of  the  total  amount  which  is  done  up  to  date. 

The  classification  is  often  a  matter  of  considerable  difference 
of  opinion,  especially  in  the  allowance  for  "  loose  rock."  All 
boulders,  etc.,  exceeding  the  limit  for  loose  rock  must  be 
carefully  measured.  When  there  is  much  of  this  to  do,  a  good 
plan  is  to  have  a  man  especially  to  look  after  it  on  two  or  three 
subdivisions,  who  can  also  take  the  Force  Account  and  give  to 
the  contractors  any  simple  information  they  may  require  con- 
cerning the  work.  The  subdivision  engineers  and  their  men 
are  thus  saved  a  very  considerable  amount  of  time  and  work. 


TIMBER-WORK. 

137.  Timber  is  usually  measured  in  railroad  structures  in 
B.M.  (Board  Measure),  the  contract  for  culverts,  etc.,  being 
let  by  the  1000  feet  B.  M.  One  foot  B.  M.  =  144  cubic  inches, 
so  that  the  B.  M.  of  any  given  stick  is  found  by  multiplying 
together  the  width  and  thickness  in  inches  and  the  length  in 
feet,  and  dividing  the  result  by  12. 

The  first  portion  of  this  calculation  and  the  division  by  12  is 
accomplished  by  means  of  the  table  on  page  151. 

In  altering  the  length  of  trestle-posts,  etc.,  to  make  allow- 
ance for  the  difference  in  elevation  of  the  two  rails,  the  follow- 
ing table  will  be  found  useful,  as  well  as  in  many  similar 
operations; 


RAILROAD   CONSTRUCTION". 


153 


FRACTIONS  OF  AN  INCH  IN  DECIMALS  OF  A  FOOT. 


In. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

0 

Foot  .0833 

.1667 

.2500  .3333 

.4167 

.5000 

.5833 

.6667 

.7500 

.8333 

.9167 

.0026  .0859 

.1693 

.2526  .3359 

.4193 

.5026 

.5859 

.6693 

.7526 

.8359 

.9193 

3i 

.0052  .0885 

.1719 

2552'  3385 

.i219 

.5052 

.5885 

.6719 

.7552 

.8385 

.92191 

A 

.0078  .0911 

.1745 

.2578 

.3411 

.4245 

.5078 

.5911 

.6745 

7578 

.8411 

.9245 

A 

0104  .0938 

.1771 

.2604 

.343S 

.4271 

.5104 

.5938 

.6771 

.7604 

.8438 

9271 

& 

.0130  .0964 

.1797 

.2630 

.3464 

.4297 

.5130 

.5964 

.6797 

.7630 

.8464 

.9297 

TB 

.0156  .0990 

.1823 

.2656 

.3490 

4323 

.515G 

.5990 

.6823 

7656 

.8490 

.9323 

& 

.01821.1016 

.1849 

.2682 

.3516 

.4349 

.5182 

.6016 

6849 

.7682 

.8516 

.9349 

1 

.0-208  '.1042 

.1875 

.2708 

.3542 

.4375 

.5208 

.6042 

.6875 

.7708 

.8542 

9375 

i 

.0234:.  1068 

.1901 

.2734 

.3568 

.4401 

.5234 

.6068 

.6901 

7734 

.8568 

.9401 

6 

0260  .1094 

.1927 

.2760 

.3594 

.4427 

.f.260  .6094 

6927 

.7760 

.8594 

.9427 

l| 

.0286  .1120 

.1953 

.27861.3620 

.4453 

.5286  .6120 

.6953 

.7786 

.8620 

.9453 

3 

0313  .1146 

.1979 

2813  .3646 

.4479 

.5313  .6146 

.6979 

.7813 

.8646 

.9479 

13 

.0339 

.1172 

.2005 

.2839 

.3672 

.4505 

.5339 

.6172 

.7005 

.7839 

.8672 

.9505 

_72 

.0365 

.1198 

.2031 

2865 

.3698 

.4531 

.5365 

.6198 

.7031 

.7865 

.8698 

.9531 

|| 

.0391 

.1224 

.2057 

.2891 

.3724 

.4557 

.5391 

.6224 

.7057 

.7891 

.8724 

.9557 

! 

.0417 

.1250 

.2083 

2917 

.3750 

.4583 

.5417 

.6250 

.7083 

.7917 

.8750 

.9583 

ft 

.0443 

.1276 

.2109 

.2943 

.3776 

4609 

.5443 

6276 

.7109 

.7943 

.8776 

.9609 

.0469 

.1302 

.2135 

.2969 

.3802 

4635 

.5469 

6302 

.7135 

.7969 

.6802 

.9635 

ft 

.0495 

.1328 

.2161 

.2995 

.3828 

.4(561 

.5495 

.6328 

.7161 

.7995 

.8828 

.9661 

I 

.0521 

.1354 

.2188 

3021 

.3854 

.4688 

.5521 

6354 

7188 

.8021 

.8854 

.9688 

3i 

.0547 

.1380 

.2214 

.3047 

.3880 

.4714 

.5547 

.6380 

7214 

.8047  .8880 

.9714 

IB 

.0573 

.1406 

.2240 

.3073 

.3906 

4740 

.5573 

.6406 

.  7240 

.80731.8906 

.9740 

II 

.0599 

.1432 

.2266 

.3099 

.3932 

.4766 

.5599 

.6432 

.7266 

8099 

.8932 

.9766 

3 

.0625  .1458 

.2292 

.3125 

.3958 

.4792 

.5625  6458 

.7292 

.8125 

.8958 

.9792 

25 

.0651  .1484 

.2318 

.3151 

.3984 

.4818 

.56511.6484 

.7318 

81  51  !.  8984 

.9818 

13 

.0677 

.1510 

.2344 

.3177 

.4010 

.4844 

.5677 

.6510 

7344 

.8-177 

.9010 

9844 

§5  • 

.0703 

.1536 

.2370 

.3203 

.4036 

.4870 

.5703  6536 

.7370 

.8203  .9036 

.9870 

7 

.0729  .1563 

.2396 

.3229 

.4063 

.4896 

.5729  6563 

.7896 

.8229  .9063 

.9896 

§5 

.0755  .1589 

.2422 

.3255 

.4089 

.  4922 

.5755  ,.6589  .7422 

.8255  .9089 

.9922 

15 

.0781 

.1615 

.2448 

.3281 

.4115 

.4948 

.5781 

6615  .7448 

.  8281  ;.  91  15 

.91148 

Is 

.0807 

.1641 

.2474 

.3307 

.4141 

4974 

.5807  .6641 

.7474 

.8307  .9141 

.9974 

0 

1 

2 

3 

4 

5 

6 

7 

8  ' 

9 

10 

11 

For  notes  on  the  strength,  etc.,  of  timber,  see  Part  IV. 

IRON-WORK, 
138.    In  estimating  the  weight  of  Bolts  and  Nuts  the 

weight  of  the  heads  and  nuts  themselves  may  be  taken  from 
the  following  table,  assuming  them  to  be  of  ordinary  propor- 
tion : 


WEIGHT  OF  BOLT-HEAD  AND  NUT. 

Diameter  of  Bolt. 

i 

1 

\ 

I 

Y 

I 

1 

11 

1* 

If 

2 

2* 

Hex.    Head   and 

Ibs. 

Ibs 

Ibs. 

Ibs 

Ibs. 

Ibs 

Ibs 

Ibs. 

Ibs. 

Ibs. 

Ibs 

Ibs 

Nut  

.017 

.057 

.128 

.27 

.43 

.73 

1,1 

22 

38 

56 

8  8 

17 

Sq.     Head     and 
Nut  

.021 

.069 

.164 

.32 

.55 

.88 

1.3 

2.6 

4.4 

7.0 

10.5 

21 

154 


RAILROAD    CONSTRUCTION. 


The  weight  of  the  shanks  of  the  bolts  may  be  found  from 
the  following  table  of  the  weight  and  strength  of  iron  rods. 
If,  however,  the  screw  end  is  upset,  with  a  consequent  enlarge- 
ment of  the  nut  and  head,  the  usual  allowance  for  the  weight 
due  to  upsetting,  and  square  head  and  nut,  will  be  equal  to 
about  13  diameters  of  additional  length  of  the  shank  of  the 
bolt.  If  the  nut  and  head  are  hexagonal,  11  diameters  are  then 
sufficient.  This  allowance  is  suitable  when  the  length  of  the 
upsetting  equals  about  6  diameters  of  the  shank.  Thus  if  we 
have  a  1-inch  bolt  upset  for  6",  if  36"  long  and  the  head  and 
nut  square,  its  weight  will  be  given  by  the  weight  of  a  1-inch 
bar  49 "  long. 


WEIGHT  AND  STRENGTH   OF  ROUND  WROUGHT- 
IRON  BARS. 


Diam. 
in 
inches. 

Weight  in 
Ibs.  per  foot 
run. 

Breaking 
Strain  in 
Ibs. 

Diam. 
in 
inches. 

Weight  in 
Ibs.  per  foot 
run. 

Breaking 
Strain  in 
Ibs. 

1 

.0414 

550 

II 

3.35 

42310 

A 

093 

1240 

1; 

4.13 

5-3200 

{ 

.165 

2200 

1| 

5.00 

63170 

T5« 

.258 

3430 

1, 

5.95 

76260 

3 

.372 

4950 

1- 

6.99 

88260 

.506 

6720 

*l 

8.10 

102370 

.661 

8800 

tj 

9,30 

117600 

i 

.837 

11130 

2 

10  6 

133700 

1.03 

18750 

2; 

\ 

12.0 

142900 

k 

1.25 

16620 

^ 

13  4 

160400 

1.49 

19780 

g' 

14  9 

178500 

}! 

1.75 

23300 

2: 

• 

16.5 

198000 

2.03 

26880 

i 

18.2 

218200 

i 

2.33 

30910 

% 

20.0 

239100 

1 

2.65 

35170 

3 

23.8 

285000 

As  a  safe  working  strain  one  fifth  of  the  above  breaking  strains 
may  usually  be  taken. 

The  two  washers  generally  used  to  each  bolt  weigh  together 
about  the  same  as  a  length  of  shank  =  14  diameters;  but  if  the 
bolt  is  upset,  they  then  weigh  about  the  same  as  a  length  =  22 
diameters. 

Iiiiilroiid  Spikes. — The  following  table  gives  the  weight, 
etc. ,  of  the  spikes  commonly  used  for  fastening  the  rails  to  the 
ties: 


RAILROAD   CONSTRUCTION. 


155 


Length 
in 
inches. 

Thick- 
ness in 
inches 

No.  pet- 
keg  of 
150  Ibs. 

No.  per 
Ib. 

Length 
in 
inches. 

Thick- 
ness in 
inches. 

No.  per 
keg  of 
150  Ibs. 

No.  per 

4I 

. 

400 

2.66 

& 

I 

350 

2.33 

5 

3 

705 

4.70 

5£ 

d£ 

289 

1.93 

5 

_7g 

488 

3.25 

4 

•> 

218 

1.46 

5 

X 

390 

2.60 

6 

1 

310 

2.07 

5 

9 

295 

1»  97 

6 

JL 

262 

1.75 

5 

1 

257 

1.71 

6 

B 

196 

1.30 

The  following  table  gives  the  angle-bars  and  bolts  neces- 
sary lor  1  mile  of  track : 


Length  of 
Rails 
in  feet. 

No.  of 
Angle- 
bars. 

No.  of 
Bolts. 

Length  of 
Rails 
in  feet. 

No.  of 
Angle- 
bars. 

No.  of 
Bolts. 

24 

880 

1760 

27 

782 

1564 

25 

844 

1688 

28 

754 

1508 

26 

812 

1624 

30 

704 

1408 

The  following  table  gives  the  weight  of  Rails  required  for 
1  mile  of  track  : 


Weight 
of  Rail 
per  yard. 

Weight 
per  mile. 

Weight 
of  Rail 
per  yard. 

Weight 
per  mile. 

Weight 
of  Rail 
per  yard. 

Weight 
per  mile. 

Ibs. 

tons.  Ibs. 

Ibs. 

tons.  Ibs. 

Ibs. 

tons.     Ibs. 

40 

62      1920 

56 

88           0 

65 

102       320    . 

45 

70      1600 

57 

89      1280 

68 

106      1920 

48 

75        960 

60 

94        640 

70 

110           0 

50 

78      1280 

62 

97        960 

72 

113        820 

52 

81       1600 

64 

100       1280 

76 

119        960 

The  weight  of  iron  required  per  mile  is  very  nearly  given 
by  the  rule:  Multiply  the  weight  in  Ibs.  per  yard  by  If;  the 
product  is  the  weight  required  in  tons  of  2000  Ibs.  (the  tons 
in  the  table  =  2240  Ibs.) 

The  weight  of  iron  in  Ibs.  per  yard  is  given  by  multiplying 
its  sectional  area  in  inches  by  10,  assuming  the  iron  to  weigh 
480  Ibs.  per  cubic  foot.  Steel  rails  usually  weigh  about  490 
Ibs.  per  cubic  foot. 


156 


RAILROAD   CONSTRUCTION. 


130.  BALLAST  AND  TIES.— The  following  table  gives 
the  amount  of  ballast  required  per  mile  of  road: 


Depth 

Top  Width,  Single  Track. 

Top  Width,  Double  Track. 

inches. 

10  Ft. 

11  Ft. 

12  Ft. 

21  Ft. 

22  Ft. 

23  Ft. 

cu.  yds. 

cu.  yds. 

cu.  yds. 

cu.  yds. 

cu.  yds. 

cu.  yds. 

12 

2152 

2347 

2543 

4303 

4499 

4695 

18 

3374 

3667 

3960 

6600 

6894 

7188 

24 

4694 

5085 

5474 

8996 

9388 

9780 

30 

6111 

6600 

7087 

11490 

11980 

1:3470 

This  table  assumes  that  the  side-slopes  of  the  ballast  are  at 
the  rate  of  1  to  1 ,  and  that  there  is  a  space  of  6  feet  clear  be- 
tween the  tracks. 

The  following  table  gives  the  number  of  Ties  required 
per  mile  of  track : 


Centre  to  Centre 
in  inches. 

No.  of  Ties. 

Centre  to  Centre 
in  inches. 

No.  of  Ties. 

18 

3520 

27 

2347 

20 

3168 

30 

2112 

22 

2889 

33 

19-20 

24 

2640 

36 

1760 

For  useful  information  in  connection  with  Construction, 
see  Part  IV. 


PART   III. 
EXPLORATORY  SURVEYING. 


140.  IN  Part  I  we  have  already  considered  the  subject  of 
"  Preliminary  Surveys,"  made  principally  with  the  object  of 
obtaining  topography  by  means  of  which  the  final  location  for 
a  railroad  may  be  selected.     We  will  here  deal  with  the  sub- 
ject of  rough  Reconnoissance  and  Exploratory  Surveys,   in 
which  accuracy — such  as  it  is  generally  understood — is  not 
essential,  and  in  which  the  general  bearings  of  rivers  and 
streams,  and  the  elevations  of  mountain  passes,  etc.,  plotted  to 
a  scale  of  a  mile  or  so  to  an  inch,  are  the  main  points  to  be 
established. 

But  before  dealing  with  the  problems  which  arise  in  explora- 
tory surveying  it  will  be  well  to  consider  the  Instruments 
usually  employed  in  this  class  of  work. 

INSTEUMENTS. 

141.  The  Instruments  generally  used  in  Reconnoissance  and 
Exploratory  Surveys  are  the  following:     The  Sextant,  Chro- 
nometer, Artificial  Horizon,  and  the  Cistern  and  Aneroid  Ba- 
rometers.    To  these  may  be  added  with  advantage,   a  light 
portable  Transit. 

We  will  treat  each  separately  in  the  order  here  given. 

The  Sextant. 

There  are  in  common  use  two  forms  of  sextant — the  Nautical 
and  the 'Box  sextant;  but  since  the  latter  is  nothing  more  than 
the  former  reduced  into  a  small  portable  shape,  we  can  con- 
sider them  both  under  one  head.  For  astronomical  work  the 

157 


158  EXPLORATORY    SURVEYING. 

box-sextant  may  be  considered  almost  worthless,  but  for  taking 
ordinary  topography  it  is  an  extremely  handy  instrument,  and 
in  more  extensive  work  it  is  a  very  useful  support  to  a  nautical 
sextant  in  many  ways.  The  ADJUSTMENTS  of  the  sextant 
are  as  follows: 

A.  To  place  the  index-glass  perpendicular  to  the  plane 
of  the  instrument.— Set  the  index  to  about  60°,  and  then, 
looking  at  the  image  of  the  limb  of  the  instrument  as  reflected 
in  the  index-glass,  the  real  limb  and  the  image  should. appear 
to  form  one  continuous  arc.     If  they  do  not  do  so,  the  index- 
glass  must  be  moved  by  means  of  the  screws  at  its  back  (see 
Fig,  75)  until  it  does. 

B.  To  place  the  horizon-glass  perpendicular  to  the 
plane  of  the  instrument. — Clamp  the  index  near  to  zero, 
and  then,  looking  at  some  well-defined  object,  turn  the  tangent- 
screw  of  the  index  until  the  object,  as  seen  directly,  and  its  re- 
flected image  are  brought,  if  possible,  to  coincide.     If  they 
cannot  be  made  to  coincide  the  horizon-glass  is  out  of  adjust- 
ment and  must  be  corrected  by  means  of  the  adjusting  screws 
with  which  it  is  fitted. 

C.  To  obtain  the  index-error. — For  the  purpose  of  meas- 
uring the  index-error  when  it  is  negative,  i.e.,  when  the  cor- 
rection for  it  is  to  be  added,  the  graduations  of  the  limb  are 
carried  a  short  distance  back  from  zero  through  what  is  termed 
the  ABC  OP  EXCESS.     The  index-error  is  obtained  by  noticing 
the  reading  when  the  coincidence  mentioned  in  Adjust.  B  is 
obtained.     But  in  this  case  the  object  must  be  a  far  distant  one, 
so  that  the  reading  may  not  be  affected  by  instrumental  paral- 
lax.    Had  the  index  been  set  exactly  at  zero  when  the  above- 
mentioned  coincidence  was  made,  there  would  of  course  be  no 
index-error,  but  it  is  usually  better  to  apply  an  index-error  than 
to  attempt  to  obtain  an  exact  coincidence  at  zero. 

A  very  accurate  method  of  obtaining  the  index-error  is  to 
measure  the  diameter  of  the  sun  several  times  "  on  and  off  the 
arc  " — i.e.,  on  the  positive  and  negative  side  of  zero:  the  mean 
of  the  readings  will  then  be  the  correction,  positive  if  on  the 
main  arc,  and  negative  if  on  the  arc  of  excess.  Thus,  for  ex- 
ample, if  the  diameter  of  the  sun  measured  on  the  main  arc  = 
32'  20",  and  on  the  arc  of  excess  30'  40",  the  mean  being  0'  50" 
on  the  main  arc,  shows  that  50"  has  to  be  subtracted  from  all 
angles  as  read  from  zero  on  the  main  arc,  i.e.,  that  the  coinci- 


EXPLORATORY   SURVEYING.  159 

dence  mentioned  in  Adjust.  B  occurs  when  the  reading  is  50" 
on  the  main  arc. 

D.  To  correct  for  eccentricity.— A  common  error  to 
which  all  sextants  are  liable  is  eccentricity  of  the  centre  of  mo- 
tion of  the  index-arm  and  the  centre  of  the  graduated  arc.  It 
unfortunately  admits  of  no  adjustment,  but  corrections  for  it 
maybe  obtained  as  follows:  "  As  it  has  no  appreciable  effect 
on  small  angles,  it  is  advisable — using  the  artificial  horizon—- 
to take  a  set  of  altitudes,  say  10,  which  will  form  a  mean  of 
about  100°  on  the  arc,  noting  the  time  of  each  accurately  by  a 
trustworthy  chronometer;  should  the  time  so  found  coincide 
with  the  known  rate  of  the  chronometer  there  is  no  error. 
Should  the  results  differ  by  several  seconds  of  time,  it  may  be 
assumed  that  the  error  of  the  instrument,  combined  with  per- 
sonal error,  has  caused  it.  By  the  rate  at  which  the  sun  was 
rising  or  going  down  during  the  observations,  the  amount  of 
angle  due  to  those  seconds  is  easily  found  (see  Sec.  195). 
Half  that  amount  will  be  .the  error  of  the  sextant  upon  that 
angle.  As  an  EXAMPLE,  suppose  by  a  morning  observation  the 
true  reflected  altitude  =  100°,  while  the  instrument  made  it 
100°  01',  the  calculation  would  make  it  about  3  seconds  later 
than  the  truth.  In  the  afternoon  a  similar  error  would  make 
it  3  seconds  earlier.  Thus  a  disagreement  of  about  6  seconds 
arises  for  about  1'  of  altitude.  By  4  or  5  such  sets  of  altitudes 
at  different  parts  of  the  arc  sufficient  data  will  be  procured 
from  which  to  form  a  table  of  corrections  for  all  altitudes." 

142.  The  sextant,  unlike  the  transit,  has  the  apex  of  the 
angle  which  it  measures  not  coincident  with  any  particular 
part  of  the  instrument,  but  varying  its  position  according  to 
the  magnitude  of  the  angle  observed.  This  is  due  to  what  is 
usually  called  Instrumental  Parallax,  and  arises  from  the 
fact  that  the  index -glass  is  not  situated  .in  the  direct  line  of 
sight.  This  may  be  best  shown  by  means  of  Fig.  75. 

Suppose  8  and  R  are  two  objects,  the  angle  between  which 
we  wish  to  measure.  When  the  index-arm  has  been  so  placed 
that  the  image  of  S -is  reflected  from  the  index-glass  7,  so  as  to 
coincide  with  R  as  seen  directly  through  the  horizon-glass  H, 
the  angle  which  is  given  by  the  sextant  is  the  angle  8AE, 
where  A  is  a  point  in  the  line  of  sight,  found  by  producing  81 
to  its  intersection.  But  suppose  8'  and  E  were  the  two  objects 
between  which  the  angle  is  to  be  observed,  then  a  will  be  the 


160  EXPLORATORY   SURVEYING. 

apex  of  the  angle  measured.  Finally,  if  8  is  situated  at  s,  so 
that  si  is  parallel  to  RA,  then  the  angle  given  by  the  sextant 
between  s  and  R=Q°  (i-e.,  if  there  were  no  index-error  the 
reading  should  be  zero),  and  if  the  reflection  of  H  were  brought 
to  coincide  with  R  as  directly  seen,  then  the  angle  observed 
would  be  negative,  and  would  thus  be  read  on  the  "  arc  of  ex- 
cess/ and  be  equivalent  to  IRA.  If  R  is  at  a  distance  from 
the  instrument  so  great  that  HI  and  RA  are  sensibly  parallel, — 
as  was  assumed  in  Adjustment  C, — the  question  of  instrumental 


FIG.  75. 

parallax  may  be  ignored;  but  in  measuring  angles  between  two 
objects  when  the  object  directly  looked  at  is  near  at/hand,  the 
instrument  must  be  either  so  placed  that  the  apex  will  coincide 
with  the  position  at  which  the  angle  is  to  be  observed,  or  else  a 
correction  applied,  the  angle  as  given  by  the  sextant — taking, 
say,  the  index-glass  as  the  constant  apex  of  the  angle— being 
always  too  small. 

In  using  an  artificial  horizon  there  is  another  form  of  paral- 
lax which  sometimes  needs  consideration  due  to  the  apex  A  of 
the  angle  observed  not  coinciding  with  the  artificial  horizon. 
Let  R  be  the  image  of  a  star  8  reflected  in  the  artificial  horizon. 
Then  if  8 A  is  parallel  to  SRt  as  is  sensibly  the  case  when  deal- 
ing with  objects  at  a  considerable  distance  from  the  instrument, 
the  angle  8AR  may  be  considered  equal  to  twice  the  angle  8RB; 
i.e.,  the  altitude  read  on  the  sextant  is  the  "  double-altitude"  of 
the  star,  which  needs  dividing  by  two  in  order  to  obtain  the 
altitude;  but  where  Sis  comparatively  close  at  hand,  then  we 
cannot  consider  8AR  =  2SRB,  and  consequently  by  dividing 


EXPLORATORY   SURVEYING.  161 

the  reading  on  the  arc  by  two,  it  is  not  the  altitude  as  reflected 
from  the  horizon  which  is  observed,  but  from  a  point  r  so 
situated  tbat  the  angle  ASr  is  equal  to  the  angle  RSr.  Suppose 
we  select  this  point  r  in  the  line  of  sight,  as  in  Fig.  75,  then  it 
may  be  easily  proved  that  if  rb  is  parallel  to  RB  (the  surface  of 
the  artifiofltl  horizon)  Srb  —  %SAR.  And  since  the  sines  of 
small  angles  may  be  assumed  to  be  proportional  to  the  angles 
themselves,  we  may  consider  the  point  r  to  be  situated  half- 
way between  A  and  R.  Thus  in  observing  an  altitude  with 
the  artificial  horizon,  where  the  distance  RA  is  appreciable 
compared  with  the  distance  8 A,  it  becomes  necessary  either  to 
apply  a  correction,  or  to  arrange  the  positions  of  the  horizon 
and  the  instrument  so  that  the  point  r  may  coincide  with  the 
apex  of  the  angle  which  it  is  wished  to  observe. 

143.  A  sextant  is  usually  only  graduated  up  to  about  140°. 
For  nautical  work  this  is  amply  sufficient,  but  where  an  arti- 
ficial horizon  is  used — since  the  angle  read  is  double  the  real 
altitude— the  altitude  will  be  limited  to  about  70°.     To  obviate 
this  difficulty,  sextants  are  often  supplied  with  a  contrivance 
which  consists  of  a  small  mirror  below  the  index  glass,  fixed  in 
such  a  position  that  when  the  index  is  at  the  mark  numbered 
180°  upon  what  is  called  the  SUPPLEMENTARY  ABC,  those  two 
mirrors  are  at  right  angles  to  each  other,  and  the  objects  whose 
images  appear  to  coincide  in  direction  really  lie  in  diametri- 
cally opposite  directions. 

144.  In  observing  angles  with   the  sextant,  when  the  two 
objects  and  the  observer's  eye  are  not  in  the  same  horizontal 
plane,  in  order  that  the  angle  measured  may  be  a  horizontal 
one,  it  becomes  necessary  either  to  arrange  matters  in  such  a 
way  that  the  angle  observed  between  the  objects  may  be  the 
horizontal  angle,  or  to  apply  a  correction  to  the  angle  ob- 
served. 

In  the  former  case  two  vertical  rods  may  be  ranged  in  line 
with  the  objects  and  the  observer's  eye,  and  the  angle  between 
them  then  measured  with  the  plane  of  the  sextant  horizontal. 
But  the  most  accurate  method  is  to  observe  the  angle  between 
the  objects  themselves,  and  then  to  observe  the  angle  of  altitude, 
or  depression  of  each. 

Thus,  in  Fig.  76,  let  A  and  B  be  the  two  objects,  Othe  position 
of  the  observer.  Then  if  Z  be  the  zenith  and  a  and  b  points 
where  the  vertical  planes  through  A  and  B  respectively  inter- 


162 


EXPLORATORY   SURVEYING. 


sect  the  horizontal  plane   abO,   then   Aa  and  Bb  represent 
respectively  the  altitudes  of  A  and  B,  and  the  complement  of 


&''- 


the  altitude  of  each  equals  its  "  zenith  distance,"  AZor  BZ. 
Then  in  the  spherical  triangle  ABZ,  since  we  know  all  three 
sides,  therefore  (since  ab  =  Z) 


where  8  — 


sinAZsinBZ 


145.  Every  possible  means  should  be  taken  in  observing 
angles  with  a  sextant  to  eliminate    instrumental  errors.     In 
order  to  do  this  all  careful  observations  should  be  in  "  doubles:" 
thus  if  the  observation  is  for  latitude,  a  star  north  and  a  star 
south  should  be  observed;  the  errors  of  the  instrument   will 
then  affect  the  result  in  opposite  directions,  and  taking  the 
mean  of  the  results  will  eliminate  the  errors.     So  also  an  ob- 
servation for  time  should  be  taken  in  "doubles:"  namely,  a 
star  east  and  a  star  west.     Also  in  taking  Lunar  Distances  the 
sets  should  be  taken  in  "doubles,"  one  set  of  distances  to  a 
star  east  of  the  moon  and  one  to  a  star  west. 

The  Artificial  Horizon. 

146.  The  best  substance  to  use  for  an  artificial  horizon  is 
mercury,  mainly  on  account  of  its  bright  reflecting  surface. 
In  a  wind,  however,  syrup  is  better  than  mercury,  being  more 


EXPLOKATORY   SURVEYING.  163 

viscous  and  consequently  less  liable  to  be  affected  by  currents 
of  air,  but  its  reflecting  surface  is  decidedly  inferior,  Oil,  too, 
is  frequently  made  use  of.  A  sheet  of  water  on  a  still  night 
makes  a  fairly  good  horizon. 

Black  glass  horizons,  which  can  be  levelled  up  by  means  of 
adjusting  screws,  are  sometimes  used,  but  though  at  times 
more  convenient  than  a  liquid  surface  they  are  considerably 
less  reliable.  The  best  way  to  carry  mercury  is  in  an  iron 
bottle,  which  can  be  made  by  any  blacksmith  out  of  a  piece  of 
iron  pipe,  fitted  with  a  screw  stopper  in  the  cap.  Mercury 
must  be  kept  carefully  away  from  all  greasy  substances,  and 
also  from  lead,  gold,  or  silver,  with  which  it  amalgamates. 
A  glass  cover  in  the  form  of  a  triangular  prism  is  often  of  use 
in  shielding  the  horizon  from  the  wind;  but  owing  to  the  in- 
creased probability  of  error,  due  to  refraction  in  the  cover  it- 
self, it  is  to  be  avoided  when  possible.  The  mercury  can 
usually  be  protected  from  the  wind  by  placing  it  in  a  hole 
slightly  below  the  general  surface  of  the  ground,  or  by  build- 
ing up  a  sort  of  protection  around  it.  A  wooden  trough  makes 
the  best  form  of  saucer  to  hold  it  in;  copper  also  does  well. 
It  should  have  an  outlet  at  one  corner  to  facilitate  the  pouring 
back  of  the  mercury  into  the  bottle.  About  5  inches  by  3 
inches  is  a  good  size  for  the  trough.  It  should  also  be  of  about 
uniform  depth,  which  need  not  exceed  half  an  inch. 

To  PREPARE  THE  HORIZON,  pour  the  mercury  into  a  small 
chamois-leather  bag,  leaving,  however,  a  little  behind  in  the 
bottle  as  "scum,"  and  then  squeeze  it  out  gently  into  the 
trough.  The  surface  so  obtained  is  usually  as  clear  as  could 
be  wished  for,  but  if  the  trough  or  the  leather  happens  to  have 
been  a  little  dirty,  a  film  of  dust  will  sometimes  be  found  on 
the  surface.  This  can  easily  be  cleared  away  by  sweeping  it 
lightly  with  a  feather.  The  horizon  is  then  ready  for  use. 

If  a  class  cover  is  used  over  it,  the  observation  should  be 
taken  twice,  the  cover  being  turned  around  for  the  second  ob- 
servation, and  the  mean  of  the  results  taken;  in  this  way  the 
error  arising  from  the  refraction  of  the  glass  is  more  or  less 
eliminated. 

The  mercury  should  always  be  carried  as  steadily  as  possible, 
the  bottle  being  kept  "end  up." 

Altitudes  less  than  about  6°  cannot  be  read  with  the  artificial 
horizon  on  account  of  the  obliquity  of  the  rays. 


164  EXPLORATORY   SURVEYING. 

An  artificial  horizon  is  almost  always  to  be  preferred  to  a 
natural  horizon,  such  as  is  given  at  sea,  on  account  of  the 
refraction  of  the  air,  as  regards  the  horizon  itself,  not  entering 
appreciably  into  the  question. 

The  Chronometer. 

147.  Chronometers  have  been  found  by  experience,  when 
subjected  to  the  shakings  and  joltings  which  necessarily  more 
or  less  accompany  their  transportation  on  laud,  to  be  very  un- 
reliable instruments.  A  small  pocket-chronometer  is  usually 
almost  as  reliable  for  land  work  as  one  of  larger  and  finer 
make,  being  less  liable  to  derangement. 

As  regards  the  care  of  chronometers,  they  should  always  be 
kept  as  much  as  possible  in  the  same  position,  and  be  always 
wound  at  the  same  time  of  day,  and  wound  to  the  butt.  Also, 
they  must  be  kept  away  from  all  magnetic  influence,  such  as 
is  often  caused  by  their  proximity  to  iron.  They  should,  of 
course,  be  rated  before  starting  out,  but  if  they  are  new  chro- 
nometers they  will  probably  gain  on  their  "  rate. "  The  "  shop- 
rate"  is  almost  always  different  from  the  field-rate,  so  that 
really  very  little  dependence  is  to  be  placed  on  them  compared 
with  that  placed  on  chronometers  at  sea.  But  though  the  rate 
when  out  on  the  work  may  be  entirely  different  from  what  it 
was  before  starting,  yet  the  rate  in  the  field  will  be  more  or 
less  constant ;  and  though  no  great  dependence  can  be  placed 
on  the  actual  position  as  given  by  a  chronometer  after  consid- 
erable jogging  and  jolting,  yet  it  serves  to  connect  the  various 
stations  observed,  relatively  to  each  other,  with  a  fair  amount 
of  accuracy  when  the  intervals  of  time  between  the  observa- 
tions are  not  great.  These  positions  can  then  be  finally  cor- 
rected after  the  general  field-rate  of  the  chronometer  has  been 
ascertained. 

As  regards  allowing  for  temperature,  that  can  only  be  done 
by  an  actual  testing  at  different  temperatures.  Every  chro- 
nometer goes  fastest  in  some  certain  temperature  which  has  to 
be  calculated  from  the  rates  that  it  makes  at  three  fixed  tem- 
peratures; then  as  the  temperature  varies  from  that  at  which 
the  chronometer  goes  fastest,  so  its  rates  vary  in  the  ratio  of 
the  square  of  the  distance  in  degrees  of  temperature  from  its 
maximum  gaining  temperature.  A  fair  test  for  a  pocket- 


EXPLORATORY  SURVEYING.         165 

chronometer  is  to  place  it  in  four  extreme  positions  and  let  it 
stay  in  each  for  24  hours  ;  if  the  rate  for  any  position  does  not 
vary  by  more  than  rive  seconds  from  the  rate  in  any  other 
position,  the  watch  is  as  good  as  can  generally  be  found. 

BAROMETERS. 

148.  There  are  two  kinds  of  barometers  used  in  exploratory 
surveying — the  "CISTERN"  form  of  the  mercurial  barometer, 
and  the  "ANEROID." 

The  Cistern  barometer,  owing  to  its  size,  is  mainly  suitable 
for  use  in  camp  as  a  standard  with  which  the  Aneroids  may  be 
compared. 

The  nature  of  the  difficulties  involved  in  observing  the  dif- 
ference of  elevation  between  any  two  points  may  be  best 
shown  as  follows  : 


FIG.  77. 

In  Fig.  77,  suppose  we  have  two  stations,  A  and  B,  whose 
difference  in  elevation  we  wish  to  determine.  If  the  atmos- 
phere were  in  a  state  of  rest  there  would  be  no  difficulty  in 
devising  formulas  which  should  give  correct  results,  supposing 
the  instruments  themselves  recorded  correctly,  for  then  the 
barometric  reading  along  the  horizontal  line  CB  would  at  all 
points  be  the  same,  and  we  should  simply  have  to  obtain  a 
formula  founded  on  Boyle  and  Mariotte'sjaw  for  the  pressure 
of  gases,  to  obtain  the  difference  in  the  heights  of  A  and  0 
which  should  correspond  with  the  observed  difference  in 
pressure.  But  since  the  atmosphere  is  always  more  or  less 
subject  to  disturbing  influences,  such  as  temperature,  humid- 
ity, etc.,  which  cause  the  barometric  gradient  at  B  to  assume 
such  forms  as  BD  or  BE,  no  formula  founded  on  statical 
principles  can  possibly  be  expected  to  give  correct  results  ;  yet 
any  formula  which  attempts  to  take  account  of  the  fluctuations 
in  gradient  necessitates  a  knowledge  of  the  temperature, 


166  EXPLORATOEY   SURVEYING. 

humidity,  and  general  state  of  the  atmosphere  between  A  and 
B,  which  it  is  impossible  to  obtain.  By  taking  observations  at 
points  immediately  between  A  and  B  some  allowance  may  be 
made  for  these  various  disturbances,  but  as  a  rule  very  little 
is  gained  by  so  doing  compared  with  the  time  and  labor  which 
it  involves. 

Since  the  variations  in  gradient  are  generally  too  rapid  to 
allow  of  the  state  of  the  atmosphere  at  one  hour  being  of  much 
service  in  indicating  its  probable  condition  a  few  hours— or 
even  minutes — later,  it  follows  that  labor  spent  in  reducing 
barometric  readings  between  two  such  stations  as  A  and  B,  by 
applying  corrections  for  latitude  and  various  other  require- 
ments which  are  often  employed,  simply  results  in  a  mathe- 
matical illusion  which  is  possibly  erroneous  to  the  extent  of 
50  or  100  p.  c. 

The  best  way  to  proceed  in  ordinary  practice  is  to  make  use 
of  formulae  which  assume  the  air  to  be  in  a  state  of  equi- 
librium— applying  corrections  for  temperature  which  expe- 
rience has  shown  to  be  necessary— and  then  to  eliminate  the 
errors  due  to  variations  in  gradient  as  much  as  possible  by 
taking  the  mean  result  of  the  readings  on  several  occasions,  or 
by  observing  simultaneously  at  the  two  stations,  as  described 
in  Sec.  150. 

149.  The  first  information  necessary  in  devising  a  formula 
for  the  reduction  of  barometric  readings  is  the  relative  weight 
of  mercury  and  air.  This  ratio  amounts  to  about  1050,  de- 
pending upon  which  values  of  the  densities  are  employed.  The 
barometer  at  the  time  is  supposed  to  be  at  sea-level  in  latitude 
45°  at  a  temperature  of  32°  F.  This  ratio,  if  multiplied  by 
5.74 — which  is  a  factor  obtained  from  Boyle  and  Mariotte's 
law  that  the  density  of  a  gas  varies  directly  as  the  pressure  to 
which  it  is  subjected — gives  a  product  known  &$i\ie  barometric 
coefficient.  Various  values  are  given  for  this  coefficient,  but 
probably  that  given  by  Reguault  is  the  most  accurate,  namely, 
60,384  ;  from  this,  taking  no  account  of  the  effects  of  tempera- 
ture or  latitude,  we  find  that  the  difference  in  elevation  in 
feet  equals 

X=  60384  log  ^, 
where  H  is  the  barometric  reading  at  the  lower  station  and  7* 


EXPLORATORY  SURVEYING.  167 

is  the  barometric  reading  at  the  upper  station.  The  correction 
for  temperature,  as  usually  applied,  assumes  that  the  mean 
temperature  of  the  air  between  A  and  B  is  the  mean  tempera- 
ture of  the  air  at  the  two  stations.  If  we  then  take  .004  as  the 
coefficient  of  expansion  of  air  for  1°  Centigrade,  the  above 
formula  needs  multiplying  by  1  -\-  .002(77-)-  t),  where  T  and  t 
are  the  temperatures  on  the  Centigrade  scale  at  the  lower  and 
the  upper  station,  respectively  ;  and  if  we  take  Tand  t  as  the 
temperatures  on  the  Fahrenheit  scale,  then  this  factor  becomes 

T+t-M 

900 

and  this  is  usually  called  the  "  temperature  term." 

Another  factor  is  often  employed  to  correct  for  the  different 
effects  of  gravity,  due  to  difference  of  latitude.  According 
to  Laplace,  this  "  latitude  term"  equals 

1  +  .  0026  cos  2L, 

where  L  =  the  latitude.  He  also  applied  a  correction  for  the 
effect  of  altitude  above  sea-level  on  the  force  of  gravity  ;  but 
this  may  be  altogether  neglected.  A  correction  is  also  some- 
times applied  to  allow  for  the  effect  of  temperature  on  the 
barometers  themselves—  which  is  ascertained  by  having  ther- 
mometers attached  to  them.  And  since  changes  of  tempera- 
ture affect  both  the  mercury  and  the  scales  in  opposite  direc- 
tions, if  we  take  .0001  as  the  relative  expansion  of  mercury 
for  1°  F.  to  the  expansion  of  the  scales,  in  order  to  correct  the 
barometers  themselves  for  temperature,  the  above  value  of  X 
should  be  multiplied  by 


where  T'  and  t'  are  the  temperatures  as  recorded  by  the  "  at- 
tached "  thermometer  at  the  lower  and  the  upper  station, 
respectively. 
Thus  the  complete  formula  becomes 


.  0020  cos 


168  EXPLORATORY   SURVEYING. 

A  correction  for  humidity  is  sometimes  applied,  but  it 
necessitates  observations  of  the  state  of  the  air  being  taken 
with  a  hygrometer  ;  and  since  it  is  doubtful,  even  then,  whether 
any  material  advantage  is  derived  by  so  doing,  we  may  ignore 
this  correction  entirely.  We  may  simplify  the  above  equa- 
tion considerably  by  dispensing  with  the  latitude  term,  which 
in  ordinary  practice  is  never  required.  In  aneroid  barome- 
ters the  last  term  of  course  does  not  enter  into  the  question  at 
all ;  so  that  the  formula  generally  applicable  to  aneroid  barome- 
ters is 

X=  60384  log 


900 


If  ff&nd  Ti  do  not  differ  by  more  than  about  3000  feet  we 
may  do  away  with  the  logarithms  in  the  above  equation,  which 
thus  becomes,  approximately, 


H+  h  \  900       / 

The  error  involved  by  this  formula  is  inappreciable  within 
the  limits  stated. 
By  assuming  (T  -}-  1)  to  equal  108°  this  formula  becomes 

X=  55000          ». 


which  is  generally  known  as  Belville's  Formula  and  is  con- 
venient for  rough  work. 

( 


900  —    ' 

150  The  results  which  are  obtained  by  using  only  one 
barometer,  carrying  it  from  station  to  station,  are  of  course 
subject  to  all  the  errors  of  gradient  ;  and  these  errors  usually 
increase  with  the  distance  between  the  two  stations  ;  but  by 
taking  the  mean  of  several  results,  the  probable  error  becomes 
greatly  reduced.  (See  Sec.  204.)  Errors  of  gradient  may  be 
more  or  less  eliminated  by  using  TWO  BAROMETERS,  and 
observing  simultaneously  at  each  station,  the  barometers  being 


EXPLORATORY   SURVEYING. 


169 


T+t 

T+i-64 

r-f* 

T+f-64 

T+t 

T+J-64 

T+t 

T-\-  1  -  64 

900 

900 

900 

900 

20= 

-.0489° 

66° 

+  .0022° 

112° 

+  .0533° 

158° 

+.1044° 

2-2 

.0-167 

68 

.0044 

114 

.0556 

'  160 

.1067 

24 

.0444 

70 

.0067 

116 

.0578 

,  162 

.1089 

26 

.0422 

72 

.0089 

118 

.0600 

164 

.1111 

28 

.0400 

i  74 

.0111 

120 

.0622 

166 

.1133 

30 

-.0378 

!  76 

+.0133 

122 

+  .0644 

168 

+.1156 

& 

.0356 

78 

.0156 

124 

.0667 

170 

.1178 

34 

.0533 

80 

.0178 

126 

.0689 

172 

.1200 

36 

.0311 

82 

.0200 

128 

.0711 

174 

.1222 

38 

.0289 

84 

.0222 

130 

.0733 

176 

.1244 

40 

—  .0267 

86 

+  .0244 

132 

+.0756 

178 

+.1267 

42 

.0244 

88 

.0267 

134 

.0778 

180 

.1289 

44 

.0222 

90 

.0289 

136 

.0800 

182 

.1311 

46 

.0200 

92 

.U311 

138 

.0822 

184 

.1333 

48 

.0178 

94 

.0333 

140 

.0844 

186 

.1356 

50 

-.0156 

96 

+  .0356 

142 

+.0867 

188 

+.1378 

52 

.0133 

98 

.0378 

144 

.0878 

190 

.MOO 

54 

.0111 

100 

.0400 

146 

.0911 

192 

.1422 

56 

.0089 

102 

.0422 

148 

.0933 

194 

.1444 

58 

.0067 

104 

.0444 

150 

.0956 

196 

.1467 

60 

-.0044 

106 

+  .0467 

152 

+.0978 

198 

+  .1489 

62 

.0022 

108 

.0489 

154 

.1000 

200 

.1511 

64 

.0000 

110 

.0511 

156 

.1022 

202 

.1533 

compared  before  and  after  the  observations  :  and  these  errors 
may  of  course  be  still  further  reduced  by  taking  the  mean  of 
several  simultaneous  observations  ;  and  in  this  way  the  best 
results  can  probably  be  obtained.  But  between  two  stations 
there  is  usually  a  permanent  gradient  dependent  on  local 
causes,  such  as  the  topography  and  nature  of  the  ground, 
which  no  number  of  observations  would  -tend  to  eliminate, 
and  for  which  allowance  can  rarely  be  made.  It  is  largely 
due  to  this  cause  that  the  heights  of  mountains,  calculated 
from  the  mean  of  a  large  number  of  observations  which  differ 
but  little  from  each  other,  are  often  found,  when  obtained  by 
more  accurate  means,  to  be  very  largely  in  error. 

151.  There  are  two  or  three  points  in  connection  with  the 
READING    OF    BAROMETERS    that   are    worth    remembering. 
For  instance,   readings  should   never  be  taken  in  the  im- 
mediate vicinity  of  any  body  which  obstructs  the  wind.     "  If 
the  barometer  is  observed  on  the  windward  side  of  a  moun- 
tain the  reading  will  be  too  higti  ;  if  on  the  leeward  side,  too 
low."    Neither  should  readings  ever  be  taken  directly  before 
or  after  a  storm  of  wind  or  shower  of  rain,  as  the  atmosphere 
is  then  usually  in  an  unsettled  state. 

152.  "The   pressure  of  the  air  everywhere  undergoes  a 


170  EXPLORATORY   SURVEYING. 

daily  oscillation.  The  gradient  introduced  by  this  daily 
change  is  called  the  DIURNAL  GRADIENT.  The  pressure 
has  two  maxima  and  two  minima  which  are  easily  distin- 
guishable. Near  the  sea-level  the  barometer  attains  its  maxi- 
mum about  9  or  10  A.M.  In  the  afternoon  there  is  a  minimum 
about  3  to  5  P.M.  ;  it  then  rises  until  10  to  midnight,  when  it 
falls  again  until  about  4  A.M.,  and  again  rises  to  attain  its 
forenoon  maximum.  The  day  fluctuations  are  the  larger." 

"  The  annual  progress  of  the  sun  from  tropic  to  tropic 
throws  a  preponderance  of  heat  first  on  one  side  of  the  equator 
and  then  on  the  other,  which  produces  an  annual  cycle  of 
changes  in  the  pressure,  and  gives  rise  to  what  has  been  called 
the  ANNUAL  GRADIENT.  The  amount  of  this  variation 
is  quite  small,  but  increases  rapidly  toward  the  poles ;  at  the 
equator  it  rarely  exceeds  one  quarter  of  an  inch  per  year,  while 
hi  the  polar  regions  it  is  often  as  much  as  two  or  three  inches 
in  a  few  days." 

We  will  now  consider  the  barometers  themselves. 

A,  The  Cistern  Barometer. 

153.  This  is  an  awkward  instrument  to  carry  about,  but 
its  usefulness  on  exploratory  work  usually  fully  makes  up  for 
the  inconvenience  which  it  causes.     It  is  found  by  experience 
to  be  absolutely  necessary  in  carrying  forward  an  extended 
system  of  barometric  observations  to  have  at  hand  a  standard 
barometer  with  which  the  aneroids  may  be  from  time  to  time 
compared. 

A  supply  of  tubes  and  mercury  should  accompany  the 
barometer  in  case  of  accident,  and  it  should  be  provided 
with  a  wooden  and  leather  case.  When  moved  from  one 
place  to  another,  even  across  the  room,  it  should  be  screwed  up 
so  that  the  tube  and  cistern  will  be  perfectly  full,  and  gently 
turned  over,  end  for  end,  so  that  the  cistern  will  be  upper- 
most. In  wheeled  vehicles  it  should  be  carried  by  hand,  and 
on  horseback  strapped  across  the  rider's  shoulder.  By  car- 
rying it  with  the  cistern  uppermost  any  particles  of  air  which 
may  be  contained  in  the  mercury  become  disengaged  by  the 
jolting,  and  escape  at  the  end  where  they  do  no  harm. 

154.  TO  FILL  A  BAROMETER,  should  it  become  neces- 
sary to  do  so  in  the  field,  proceed  as  follows  :    Warm  both  the 


EXPLORATORY   SURVEYING.  171 

mercury  and  tube  aiid  filter  in  through  a  paper  funnel— the 
hole  of  which  does  not  exceed  ^  of  an  inch— to  about  i  of  an 
inch  from  the  top.  Close  the  end  and  turn  the  tube  on  its 
side  ;  the  mercury  will  then  form  a  bubble  which  can  be  made 
to  travel  from  end  to  end  and  gather  all  the  small  air-bubbles 
visible  that  adhered  to  the  inside  of  the  tube  while  filling. 
Let  the  bubble  pass  to  the  open  end,  fill  up  with  mercury  and 
close  the  tube.  Reverse  the  tube  over  a  basin,  when,  by 
slightly  relieving  the  pressure  against  the  end,  some  of  the 
mercury  will  be  forced  out,  forming  a  vacuum  above,  which 
ought  not  to  exceed  half  an  inch.  Close  up  again  tightly  and 
let  this  vacuum-bubble  traverse  the  length  of  the  tube  as  be- 
fore, on  the  several  sides,  absorbing  the  minute  portions  of  air 
still  left,  now  greatly  expanded  by  the  reduction  in  pressure. 
Perfect  freedom  from  air  can  be  detected  by  the  sharp  con- 
cussion with  which  the  mercury  beats  against  the  sealed  end, 
when,  with  a  large  vacuum-bubble,  the  horizontally  held  tube 
is  slightly  moved.  Any  air  which  may  still  be  left— which 
will  probably  not  affect  the  reading  by  more  than  a  few 
thousandths  of  an  inch — will  soon  escape  if  the  barometer  is 
carried  about  cistern  uppermost. 

Filling  by  boiling  is  a  slightly  more  efficient  method,  but  it 
is  a  much  more  difficult  proceeding. 

155.  IN  READING  THE  BAROMETER,  first  of  all  note  the 
temperature  on  Jhe  attached  thermometer,  then  screw  up  the 
mercury  in  the  cistern  so  that  its  surface  just  touches  the  ivory 
point,    being  careful   that    the  barometer   hangs  vertically. 
Give  a  gentle  tap  near  the  top  of  the  mercurial  column  to 
destroy  the  adhesion  of  the   mercury.     Set   the  vernier   by 
bringing  its  front  and  back  edges  into  the  same  horizontal 
plane  with  the  top  of  the  mercury  ;  then  read. 

156.  Should   the  mercury  in  the  cistern   become  so  dirty 
that  neither  the  ivory  point  nor  its  reflection  in  the  mercury 
can  be  seen,  the  instrument  must  be  taken  apart  and  cleaned. 
To  do  this  "  screw  up  the  adjusting  screw  at  the  bottom  until 
the  mercury  entirely  fills  the  tube,  carefully  invert,  place  the 
instrument  firmly  in  an  upright  position,  unscrew  and  take  off 
the  brass  casing  which  encloses  the  wooden  and  leather  parts 
of  the  cistern.     Remove  the  screws  and  lift  off  the  upper 
wooden  piece  to  which  the  bag  is  attached  ;  the  mercury  will 
then  be  exposed.     By  then  inclining  the  instrument  a  little,  a 


172  EXPLORATORY   SURVEYING. 

portion  of  the  mercury  in  the  cistern  may  be  poured  out  into 
a  clean  vessel  at  hand  to  receive  it,  when  the  end  of  the  tube 
will  be  exposed.  This  is  to  be  closed  by  the  gloved  hand, 
when  the  instrument  can  be  inverted,  the  cistern  emptied,  and 
the  tube  brought  again  to  the  upright  position.  Great  care 
must  be  taken  not  to  permit  any  mercury  to  pass  out  of  the 
tube.  The  long  screws  which  fasten  the  glass  portion  of  the 
cistern  to  the  other  parts  can  then  be  taken  off,  the  various 
parts  wiped  with  a  clean  cloth  and  restored  to  their  former 
position."  Everything  used  in  the  operation  must  be  clean 
and  dry,  and  all  breathing  on  the  parts  avoided  as  much  as 
possible. 

If  the  mercury  is  dusty  or  dimmed  by  oxide  it  may  be 
cleaned  by  filtering  through  chamois  leather,  but  if  chemically 
impure  it  must  be  rejected  and  fresh  mercury  substituted. 
The  cistern  should  then  be  filled  as  nearly  as  possible  and  the 
wooden  portion  put  together  and  fastened.  The  screw  at  the 
bottom  of  the  instrument  should  then  be  screwed  up.  "  The 
instrument  can  then  be  inverted,  hung  up  and  readjusted. 
The  tube  and  its  contents  having  been  undisturbed,  the 
instrument  should  read  the  same  as  before." 

B.  The  Aneroid  Barometer, 

157.  The  "Aneroid"  is  a  valuable  instrument  for  engin- 
eering and  exploratory  purposes  on  account  of  its  portability, 
and  though  not  to  be  compared  in  accuracy  with  the  mercu- 
rial barometer,  the  results  given  by  it  will  often  not  differ  from 
those  given  by  the  latter  sufficiently  to  be  of  importance.  It 
is  in  such  cases  as  these  that  the  aneroid  is  eminently  useful. 
But  it  is  too  liable  to  derangement,  and  subject  to  too  many 
defects,  to  warrant  its  being  used  in  any  other  way  than  to 
supplement  some  more  accurate  form  of  obtaining  elevations. 
In  dealing  with  the  mercurial  barometer,  after  the  correction 
for  temperature  has  been  applied,  the  instrumental  errors 
which  need  correcting  are  very  small ;  but  with  an  aneroid 
the  same  cannot  be  said.  Most  of  the  better  class  of  aneroids 
are  supposed  to  compensate  automatically  for  changes  in 
temperature.  This  compensation  should  be  tested  by  com- 
parison at  different  temperatures  with  a  standard  barometer, 
and  the  errors  tabulated  and  kept  for  future  reference. 


EXPLORATORY    SURVEYING. 


173 


While  reading,  Hie  aneroid  should  always  be  held  horizon- 
tally, for  the  weight  of  the  parts  themselves  has  a  very 
considerable  influence  on  the  readings:  a  difference  correspoud- 
ing'to  fifty  feet  being  not  uncommon  when  held  in  different 
positions.  The  aneroid  may  be  adjusted  by  means  of  the 
small  screw  at  its  back,  so  as  to  agree  with  the  reading  of  a 
standard  barometer,  but  when  the  difference  is  only  slight  it 
is  better  to  regard  it  as  an  ''index  error,"  and  correct  in  that 
way,  than  to  alter  the  reading. 

158.  Cheap  aneroids  commonly  have  the  SCALE  of  inches 
subdivided  so  as  to  read  the  elevations  above  sea-level.  This 
would  be  very  convenient  if  only  the  corresponding  pressure 
at  the  sea-level  were  always  the  same  as  given  on  the  index 
and  the  atmosphere  always  in  a  state  of  equilibrium.  The 
pressure  at  the  sea-level  is  generally  assumed  as  being  equiv- 
alent to  30  inches. 

Another  method  which  is  convenient,  though  "  unscientific 
and  inaccurate,"  is  that  of  having  a  movable  scale  of  elevations 
which  can  be  set  to  agree  with  the  barometer  reading  at  any 
known  elevation.  But  the  best  way  to  obtain  a  reading  is  to 
observe  the  reading  in  inches,  and  then  to  reduce  it  by  one  of 
the  formulae  already  given. 

BAROMETRIC  AND  ATMOSPHERIC  HEIGHTS. 


Bar. 
in. 

Alt'de 
feet. 

Bar. 
in. 

Alt'de. 
feet. 

Bar. 
in. 

Alt'de. 
feet. 

Bar. 
in. 

Alt'de. 
feet. 

Bar. 
in. 

AltMe. 
feet. 

21. 

9900.1 

23. 

7375  1 

25. 

5060.6 

27. 

2924.4 

29. 

940.9 

.1  9768.3 

.1 

7254.7 

.1 

4949.8 

.1 

2821.8 

.1 

845.4 

.2  9637.1 

.2  7134.7 

.2 

4839.5 

.2 

2719.6 

.2 

750.2 

.3  9506.5 

.3!  7015.3 

.3 

4729.6 

.3 

2617.8 

.3 

655.3 

.4'  9376.4 

.4'  6896.5 

.4 

4620.1 

.4 

2516.3 

.4 

56071 

.5  9-247.0 

.5  6778.1 

.5 

4511.0 

.5 

2415.2 

.5 

466.5 

.6  9118.3 

.6  6660.2 

.6 

44023 

.6 

2314.4 

.6 

372.6 

.7  8990.0 

.7  6542.8 

.7 

4294.0 

.7 

2214.0  | 

19 

279.0 

.8;  8862.4 

.8  6426.0 

.8 

4186.3 

.8 

2114.0  ; 

'.8 

185.7 

.9!  8735.3 

.9  6309.6 

Cj 

40789 

g 

2014.3 

.9 

92.7 

22.   8608.9 

24. 

6193.8 

26. 

3971.9 

2S. 

1915.0 

3O. 

0.0000 

.1  8483.0 

.1|  6078.3 

.1 

3865.4 

.1 

1816.0  i 

.1 

—  92.5 

.•->  8357.7 

.2  5963.4 

.2 

3759.3 

a 

1717.4  I 

.2 

-  184  7 

.3  8233.0 

.3  5848.9 

.3 

3653.6 

*£ 

1619.2  i 

.?. 

-  276.6 

.4  8108.7 

.4 

5734.9 

^ 

&54S.3 

^ 

1521.3 

'A 

-  368.2 

.5j  7985.1 

.5 

5621.4 

'.5 

3443.4 

ft 

1423.7 

£ 

-  459.5 

.6  7862.0 

.6 

5508.3 

.6 

33388 

'.6 

1326.5 

'.'c 

-  550.6 

.7 

7739.4 

.7 

5395.7 

t"j 

3234.6 

ft 

1229.6  i 

"7 

-  6414 

.8  7617.5 

.8  52S3.6 

'.8 

3130.8 

'.8 

1133.0 

'.8 

-  731.9 

.9 

7495.9 

.9 

5171.9 

( 

3027.4 

.9 

1036.8 

.9 

-  822.2 

174  EXPLOKATOHY    SURVEYING. 

No  advantage  seems  to  be  gained  by  the  use  of  large  ane- 
roids; in  fact  experience  shows  that  when  the  barometer  is 
subjected  to  much  shaking,  the  best  work  is  usually  done  by 
instruments  not  exceeding  3  inches  in  diameter.  The  eleva- 
tions according  to  which  the  elevation-scales  on  aneroids  are 
usually  divided  are  as  given  on  the  preceding  page,  and  are  ob- 
tained by  a  formula  similar  to  those  already  given,  assuming 
the  temperature  to  be  60°  Fahr. 

Many  scales,  however,  adopt  a  temperature  of  32°  F.,  in 
which  case  the  corresponding  elevations  will  be  reduced  in 
the  proportion  of  1.058  to  1. 

The  uncertainty  which  is  connected  with  barometric  obser- 
vations is  greatly  dependent  on  the  latitude  ;  the  barometric 
pressure  being  very  much  more  regular  in  the  tropics  than  in 
the  polar  regions. 


EXPLORATORY  SURVEYS. 

159.  There  are  three  distinct  ways  in  which  exploratory 
surveys  may  be  carried  on  : 

A.  By  a  series  of  triangulations. 

B.  By  direct  measurement  and  compass  courses. 

C.  By  astronomical  observations. 

And  though  usually  an  explorer  makes  use  more  or  less  of 
all  three  methods,  it  will  be  better  for  the  sake  of  clearness  to 
consider  each  separately. 

A.  By  a  Series  of  Triangulations. 

The  method  of  triangulating  is  mainly  suitable  to  moun- 
tainous country,  or  at  any  rate  to  country  where  a  view  of 
distant  mountain-peaks  is  to  be  had. 

Before,  however,  considering  the  practical  working  of  this 
system,  it  will  be  well  to  deal  with  a  few  of  the  principal 
trigonometrical  problems  which  arise  in  work  of  this  sort. 

In  Sec.  59  we  have  already  dealt  with  some  of  the  simpler 
forms  of  triangulation,  suitable  in  cases  where  a  straight  line 
has  to  be  continued  over  an  inaccessible  surface  ;  but  we  will 
here  consider  the  cases  of  obtaining  distances  and  directions 
of  points  relatively  to  each  other. 


EXPLORATORY  SURVEYING.          175 

160.  Given  two  inaccessible  points  A  and  B,  to  find 
their  distance  apart  and  bearing:  relatively  to  each 
other.— In  Fig.  78  let  CD  be  a  line  the  length  and  bearing  of 


which  are  known.  Observe  the  angles  ACD,  BCD,  ADC,  and 
BDC.  Then  in  the  triangle  CDA  we  have  the  angles  at  C  and 
D  and  the  length  CD,  and  can  thus  find  CA.  Similarly  in 
the  triangle  CBD  we  can  find  CB.  Then  in  the  triangle  CAB 
we  have  the  side  CA  and  CB  and  the  angle  at  C,  from  which 
we  can  obtain  the  distance  AB  and  its  bearing  relatively  to  CD. 
The  following  equations,  however,  reduce  the  work  which 
the  direct  solution  given  above  involves.  Find  an  angle  K 
such  that 

sin  ADC  sin  CBD 

t*in   7t^ -- * 

~  sin  CAD  sin  BDC  ' 
then 


f  CAB  -  ABCf\  ACB 

tan  f 3 J  =  tan  (45°  -  K)  cot  — —  ; 


CAB  +  ABC  .   CAB -ABC 
1 . 


sin  BDC  sin  ACB 
-  </^  - 


then 


and 


If  C  can  be  ranged  in  line  with  A  and  B  we  can  then  find 
the  position  of  A  and  B  separately,  as  shown  in  Sec.  59;  the 
difference  of  the  distances  so  obtained  gives  the  length  of  AB, 
and  the  bearing  is  obtained  by  direct  observation. 


176 


EXPLORATORY   SURVEYING. 


Suppose,  however,  that  in  Fig.  78  the  length  and  direction  of 
AB  is  known,  and  it  is  the  distance  CD  which  is  required. 

Then  observe  the  angles  at  C  and  D  and  obtain  CAB  as  be- 
fore, but  in  this  case  the  last  formula  becomes 

sin  CBD  sin  CAB 
sinBDCsinACB' 

This  might  be  also  solved  by  assuming  a  certain  length  for 
CD,  and  from  it  finding  as  above  what  the  length  of  AB 
must  be;  then  the  true  AB  is  to  the  value  of  AB  so  obtained 
as  the  true  CD  is  to  the  assumed  value  of  CD. 


FIG.  79. 

If,  as  in  Fig.  79,  the  lines  AB  and  CD  cross  each  other, 
the  above  formulae  apply  equally  well. 

161.  The  problem  known  commonly  as  the  "  Three-point 
Problem"  is  probably  the  most  useful  method  there  is  of 
establishing  the  position  of  any  given  point;  it  is  as  follows: 


FIG.  80. 


Suppose,  as  in  Fig.  80,  we  know  the  position  of  three  points 
A,  B,  and  C  and  wish  to  fix  the  position  of  the  point  8;  we 
<::m  do  it  by  simply  observing  the  angle  ASB  and  BSC. 


EXPLORATORY   SURVEYING.  177 

Then,  in  order  to  obtain  the  position  of  8  geometrically  ,  pro- 
ceed as  follows: 

Find  I),  the  centre  of  the  circle  ABS  (by  setting  off  at  A 
and  B  angles  equal  to  90°  —  A.SB}.  Then  draw  the  circle 
through  the  points  A,  B,  and  8.  Similarly  find  the  centre  E 
and  draw  the  circle  BCS.  Then  8,  the  point  of  intersection 
opposite  B,  is  the  position  required. 

When  one  of  the  angles  is  obtuse,  set  off  its  difference  from 
90°  on  the  opposite  side  of  the  line  joining  the  two  objects  to 
that  on  which  the  point  of  observation  lies. 

When  the  angle  ABC  =  the  supplement  of  the  sum  of  the 
two  angles,  the  position  of  8  will  be  indeterminate  by  this 
method. 

8  may  often  be  obtained  with  sufficient  accuracy  instrument- 
ally  by  plotting  the  angles  ASBaud.  BSCon  a  piece  of  tracing- 
cloth,  and  sliding  it  over  the  plan  until  the  required  position 
is  obtained.  The  "station-pointer"  is  an  instrument  much 
used  for  this  purpose,  especially  in  hydrographers'  offices, 
where  soundings  are  usually  plotted  in  this  way. 

If  accuracy  is  required  the  position  of  8  may  be  found  ana- 
lytically thus,  as  given  by  Prof.  Gillespie  : 

Let  AB  =  c;BC  =  a;  ABC  =  B;  ASB  =  8;  and  B8C  =  8'. 

Also  make  T  =  360°  -  8  -  8'  -  B, 
and  let       BA8  =  U,     and    BCS  =  V. 
Then 

«*  0  =  cot  r 


V=T-  U- 

7?      c  sin  U  „      a  sin  V 

B=-'    °r    SB  =  ' 


sm  8  sin  8' 


Thus  if         ^£#  =  33°  45', 

AB  =  6000  ft.      and      BC  =  4000  ft., 
we  find  ABC  =  104°  28'  39". 

Then  if  =  105°  08'  10"; 

whence  V=    94°  08'  11". 

8B  =  10425.1  ft.,    8A  =  7101.9  ft,     and    80  =  9342.9  ft. 


178 


EXPLO BATO R Y    SURVEYING. 


162.  The  position  of  a  point  may  also  be  fixed  by  observing 
the  bearings  from  it  of  two  known  points,  and  may  be 

found  on  the  plan  by  drawing  through  those  points  the  bear- 
ings so  obtained;  their  intersection  gives  the  point  required. 

163.  Another  common  method  of  fixing  the  positions  of 


FIG.  81. 

outlying  points  is  by  intersection,  as  in  Fig.  81,  the  position 
of  the  two  points  of  observation  A  and  B  being  known. 

164.  While  on  the  subject  of  triangulation,  it  will  be  as 
well  to  consider  the  methods  of  obtaining  the  heights  of 
mountains  trigonometrically. 

In  the  first  place,  suppose  we  are  able,  as  in  Fig.  82,  to  ob- 


FIG.  82. 


tain  two  points  A  and  B  in  the  direction  of  C  (a  point  the 
elevation  of  which  we  wish  to  obtain)  both  at  the  same  eleva- 
tion, and  to  measure  the  distance  between  them;  then 


CD  = 


AB 

cot  CAD  -  cot  CBD ' 


If,  however,  the  two  points  cannot  be  taken  at  the  same  level, 
but  have  to  be  taken  such  as  £7aud  A,  observe  the  angle  CEA, 


EXPLORATORY    SURVEYING. 


179 


and  at  A  the  altitudes  of  C  and  E,  either  with  an  artificial 
horizon  or  with  the  vertical  arc  of  a  transit.  Then  in  the 
triangle  EAC 

AC  =  EAsin  E cosec  C, 

where  the  angle  at  C  =  the  sum  of  the  altitudes  of  E  and  G 
(taken  at  A)  —  the  angle  at  E. 

This  would  of  course  hold  equally  good  if  EA  sloped  the 
other  way,  but  then  C  —  alt.  of  C  from  A  —  alt.  of  A  from 
E  —  angle  at  E.  The  correction  for  curvature  and  refraction 
given  in  Sec.  51  must  be  added  to  the  height  as  obtained 
above. 

But  suppose  it  is  not  convenient  to  obtain  a  base  as  above 
in  the  same  direction  as  C.  Then,  as  in  Fig.  83,  measure  a 

C 


base  AB  (not  necessarily  level)  and  observe  the  angles  CAB 
and  GBA.     Then  in  the  triangle  ABC 

AC  =  AB  sin  B  cosec  C. 

Next  observe  the  altitude  of  C  from  A,  I.e.,  the  angle  CAD', 
the  a 

CD  =  AC  sin  CAD. 

To  the  height  so  obtained,  the  correction  for  curvature  and 
refraction  given  in  Sec.  51  should  be  added. 

Suppose  it  is  required  to  find  the  difference  in  elevation  of 
two  inaccessible  points,  the  simplest  way  is  to  find  the  eleva- 
tion of  each  separately,  as  above,  and  subtract  the  one  from 
the  other, 


180  EXPLORATORY   SURVEYING. 

165.  In  observing  altitudes,  the   refraction  of  the  air 

enters  so  largely  into  the  question  and  varies  so  enormously 
according  to  the  condition  of  the  atmosphere,  that  every  pre- 
caution must  be  taken  to  eliminate  the  errors  due  to  it,  where 
accurate  work  is  wanted. 


FIG.  84. 

Its  nature  is  such  that  suppose  A  and  B  are  two  stations 
visible  from  each  other,  the  line  of  sight  between  A  and  B, 
instead  of  being  straight,  follows  a  curved  course  as  shown  in 
Fig.  84,  making  the  altitude  as  observed  at  A  too  great,  by 
the  amount  F,  which  is  termed  the  "angle  of  refraction." 
Similarly  the  depression  of  A  as  observed  from  B  will  be  too 
small.  Thus  the  tendency  of  refraction  is  to  make  objects 
appear  at  a  higher  elevation  than  they  really  are;  so  that  in 
observing  altitudes  a  correction  for  refraction  should  be 
always  subtracted  from  the  apparent  altitude  to  obtain  the 
true  altitude. 

In  ordinary  work  the  corrections  given  in  Sec.  51  for  both 
curvature  and  refraction  are  sufficiently  correct.  But  for 
highly  accurate  work — on  which  this  article  does  not  treat — 
various  allowances  and  corrections  must  be  made. 

Refraction  diminishes  with  altitude  and  is  slightly  greater 
over  water  than  land.  It  is  generally  at  its  maximum  during 
the  night,  and  at  its  minimum  about  noon;  but  it  is  steadier 
in  the  night  than  in  the  day  time,  and  for  this  reason  night 
work  is  usually  as  reliable  as  work  done  during  the  day. 
About  sunrise  and  sunset  are  the  worst  times  to  observe  alti- 
tudes, for  not  only  is  refraction  then  high  in  quantity,  but  also 
extremely  variable.  A  day  with  the  sky  overcast  is  a  good 
day  on  which  to  take  an  observation.  Clear  days  are  more 
subject  to  rapid  changes  than  dull  ones.  (For  Astronomical 
Refraction,  see  Sec.  184.) 

166.  A  method  of  eliminating  to  a  great  extent  the  effect 
of  refraction  in  observing  the  difference  of  elevation  of  two 


EXPLORATORY    SURVEYING. 


181 


stations  A  and  B,  is  that  of  observing  Reciprocal  Angles. 
Thus  in  Fig.  84,  at  A,  the  altitude  of  B  should  be  observed, 
and  at  B  (when  practicable)  the  depression  of  A.  Half  the 
difference  of  these  angles  will  be  the  combined  correction, 
and  the  tangent  of  half  their  sum,  multiplied  by  the  horizon- 
tal distance  between  them,  will  give  the  difference  of  level, 
after  adding  the  correction  for  curvature  of  the  earth  given  in 
Sec.  51  This  method  assumes  that  the  coefficient  of  refrac- 
tion is  the  same  at  both  A  and  B;  therefore  the  angles  should, 
if  possible,  be  observed  simultaneously,  lest  the  refracting 
power  of  the  air  should  change  in  the  interval .  (For  the  cor- 
rection for  Refraction,  see  Sec.  51.) 

167.  To  obtain  the  height  of  a  mountain  by  the  ob- 
served depression  of  the  sea  horizon.— The  depression  of 
the  horizon,  or  as  it  is  commonly  called  at  sea  the  "Dip," 
taking  R  =  the  earth's  mean  radius  of  curvature  in  feet, 
equals  in  seconds 

D  =  206265^/?^i 


therefore 


=  63.8 

D 

63.8' 


where  H=  Height  in  feet. 

Thus,  were  it  not  for  refraction,  we  could  find  the  elevation 
of  A  (Fig.  85)  by  merely  observing 
the  dip  D.  But  D'  is  the  dip  actu- 
ally observed;  so  that,  taking  re- 
fraction into  account,  the  above 
formula  becomes 


=       (nearly), 


which  can  only  be  .depended  on  to 
give  approximate  results. 

168.  In  observing  altitudes  with  a 
sextant  and  artificial  horizon,  as  for 
instance  in.  Fig.  84,  the  altitude  of 


FIG.  85. 


B  will  be  one  half  the  altitude  read  on  the  arc,  since  it  is  the 


182  EXPLORATORY    SURVEYING. 

"  double  altitude"  that  is  actually  observed.  To  find  a  point 
(7  on  the  same  level  as  the  instrument  the  altitude  can  then  be 
measured  down  from  AB,  To  observe  the  depression  of  A 
from  B  with  a  sextant  and  artificial  horizon,  we  must  estab- 
lish some  point— as  far  off  as  possible  so  as  to  reduce  parallax 
— the  altitude  of  which  exceeds  about  6°,  and  observe  its  alti- 
tude correctly,  and  then  obtain  the  angle  between  it  and  the 
object  whose  depression  we  wish  to  find.  At  night  a  star 
may  often  be  made  use  of  for  this  purpose,  allowance  being 
made  for  its  motion.  This  method  may  also  be  employed  in 
reading  altitudes  which  would  otherwise  need  the  use  of  a 
supplementary  arc.  (See  Parallax,  Sec.  142.) 

To  read  an  altitude  or  depression  with  a  transit,  observe  the 
altitude  first  in  the  usual  way,  then  "  reverse"  and  point  the 
telescope  to  the  object  and  read  its  co-altitude;  the  mean  alti- 
tude so  obtained  is  free  from  error  due  to  the  "horizontal 
axis"  not  being  truly  perpendicular  to  the  "vertical  axis" 
of  the  instrument.  The  errors  of  graduation  and  observation 
are  also  somewhat  reduced. 

169.  It  is  essential  that  a  survey  which  consists  of  a  series 
of  triangulations  should  have  an  accurate  base  to  start  from. 
Sometimes  in  exploratory  surveys  the  distance  between  two 
mountain  peaks,  or  some  prominent  objects  near  the  point  at 
which  the  survey  starts,  is  already  known  with  sufficient  ac- 
curacy to  warrant  the  line  joining  them  being  accepted  as  a 
base,  but  more  usually  it  is  necessary  to  obtain  the  distance 
between  such  points  from  a  base  more  or  less  accurately 
measured. 

For  this  purpose  of  course  as  level  a  piece  of  ground  must 
be  obtained  as  possible,  and  as  there  is  often  difficulty  in  find- 
ing such  a  site  long  enough  for  a  base,  it  becomes  necessary 
to  start  from  a  short  base  and  then  extend  it  by  a  series  of  tri- 
angulations, the  angles  of  which  fall,  if  possible,  between  the 
limits  of  30°  and  120°. 

As  regards  the  MEASUREMENT  OF  A  BASE  for  ordinary  work 
we  can  consider  a  steel  tape,  properly  tested  at  a  given  tem- 
perature, to  be  sufficiently  accurate.  The  correction  for  tem- 
perature amounts  to  about  .000007  of  the  length  of  the  tape  for 
every  1°  Fah.  Thus  a  100-foot  tape,  tested  at  a  temperature  of 
50°  F.,  would  give  a  result  too  long  by  about  3  feet  in  2  miles 
at  a  temperature  of  90°  F. 


EXPLORATORY    SURVEYING. 


183 


Since  all  maps  are  made  on  the  assumption  that  the  linear 
measurements  are  reduced  to  the  sea-level,  in  dealing  with  high 
altitudes  the  length  of  the  base  may  be  multiplied  by 

1—  (nearly), 

where  h  =  elevation  above  sea-level,  r  =  radius  of  the  earth 
(see  Sec.  206),  in  order  to  reduce  it  to  sea-level.  But  this  is. 
a  refinement  which  is  usually  only  needed  in  work  requiring 
great  accuracy. 

1 70.  In  making  a  regular  triangulation  survey,  the  angles 
of  the  main  triangles  are  of  course  themselves  observed;  but 
in  such  work  as  exploratory  surveys,  where  mountain  peaks 
are  selected  as  "  stations,"  such  a  method  of  procedure  would, 
on  account  of  the  time  and  difficulty  involved,  be  out  of 
the  question.  A  readier  method  of  proceeding  may  be  best 
shown  by  an  example  as  in  Fig.  86.  It  depends  upon  always 
having  in  view  at  any  station  at  least  two  points  wlwse  positions 
are  known. 

Suppose  we  have  obtained,  by  triangulation  or  otherwise, 
the  distance  between  and  bearing  of  two  conspicuous  points 
A&ndB,  and  suppose  our  route  lies  along  the  dotted  line  abed. 


FIG.  86. 

At  a,  a  point  from  which  A  and  B  are  visible,  we  observe 
the  bearings  of  A  and  B,  and  thus  fix  the  position  of  a.  Sup- 
pose that  from  a  a  distant  mountain  peak  C  is  visible,  we  take 
the  bearing  of  it  also;  then  if  we  wish  to  fix  the  position  of  such 
apoiut  as  b,  from  it  we  observe  the  bearings  of  l?and  0.  When 


184  EXPLORATORY   SURVEYING. 

we  get  to  c  we  locate  its  position  by  bearings  fro  in  A  and  B; 
but  suppose  we  can  see  A  and  B  no  farther,  it  then  becomes 
necessary  to  establish  two  other  points  which  we  may  use  as 
we  have  already  used  A  and  B.  A  bearing  to  G  will  then  lo- 
cate it.  We  also  observe  the  bearing  of  D.  When  d  is 
reached,  we  observe  the  bearings  of  C,  c,  and  D,  which  fix 
its  position  and  also  the  position  of  D. 

No  simpler  way  of  keeping  a  course  can  be  had  than  this; 
and  it  has  the  enormous  advantage  over  many  of  the  methods 
in  use,  that  it  fixes  the  main  topographical  features  bordering 
along  the  route  at  the  same  time  as  positions  on  the  route  itself. 
The  explorer  must  be  constantly  on  the  lookout  for  points 
ahead  on  his  probable  route  and  in  the  neighborhood.  The 
drawback  to  the  method  is  its  inaccuracy  when  worked  by 
magnetic  bearings  alone.  But  if  the  points  are  well  selected, 
an  error  of  a  degree  or  so  in  the  bearings  is  really  immaterial  in 
work  of  this  class,  and  the  errors  usually  more  or  less  counter- 
act each  other.  Besides,  from  time  to  time  the  courses  and 
distances  can  be  easily  checked  by  the  establishment  of  another 
base,  and  the  work  already  done  more  or  less  corrected,  and  a 
fresh  start  made. 

If  we  keep  three  or  more  points  in  view  we  are  able  to  apply 
the  trigonometrical  method  given  in  Sec.  161,  and  thus  do  very 
accurate  work  so  long  as  we  are  careful  in  establishing  cor- 
rectly the  positions  of  A,  B,  0,  D,  etc. 

In  following  along  valleys,  or  in  sight  of  a  distant  range  of 
mountains,  this  method  works  admirably,  and  if  a  transit  is  at 
hand  a  check  may  be  applied  from  time  to  time  on  the  dis- 
tances and  bearings  with  very  little  trouble. 

There  is  no  need  to  apply  any  correction,  however  extensive 
the  triangulations  may  be,  for  the  curvature  of  the  earth,  since 

the  spherical  excess  of  a 
spherical  triangle  contain- 
ing 75.5  square  miles  is  only 
1";  so  that  in  a  triangle 
containing  4530  square 
miles  the  sum  of  the  three 
B  angles  only  exceeds  180° 

by  1'. 

FIG.  87.  171.    To      measure    a 

horizontal  angle  without  an  instrument  between  two 


EXPLOKATORY   SURVEYING.  185 

such  points  as  A  and  B  from  0,  as  in  Fig.  87.  Range  in  a  and 
b  with  A  and  B,  each  distant  from  0  by,  say,  50  feet.  Meas- 
ure ab,  then 

,    AOB      ab 


172.  To  measure  a  vertical  angle  without  an  instru- 
ment, probably  the  simplest  way  is  to  hold  a  pencil  vertically 
out  at  arm's  length  and  note  the  length  subtended  on  it.  Then 
if  the  distance  from  the  eye  to  the  pencil  =  I  and  p  is  the 
length  subtended  on  the  pencil, 


tan  A  = 

6 

where  A  is  the  angle  required.     Similarly  if  L  were  the  dis- 
tance of  some  object  whose  height  Hwe  wish  to  obtain, 


1  73.  Distance  across  an  open  stretch  of  water  can  often  be 
taken  with  sufficient  accuracy  by  observing  the  time  occupied 
by  the  passage  of  the  report  of  a  gun  from  one  point  to  the 
other.  This  may  be  done  in  the  day-time  if  there  is  a  tele- 
scope handy  to  watch  for  the  smoke,  but  otherwise  the  flash 
of  course  can  be  best  seen  at  night.  The  velocity  -»,  in  feet 
per  second,  with  which  sound  travels,  depends  greatly  on  the 
temperature;  thus  at  32°  F.,  «  =  1090;  at  60°  F.,  v  =  1125;  and 
at  100°  F.,  v  -  1175. 

By  taking  the  mean  of  3  or  4  shots,  the  distance  may  be 
obtained  with  confidence  to  a  quarter  of  a  mile.  If  the  wind 
is  blowing  hard  in  the  direction  from  which  the  sound  comes, 
the  velocity  of  the  wind  may  be  added  tcrv. 

174.  We  can  observe  an  interval  of  time  when  a  watch 
is  not  at  hand  by  counting  the  vibrations  of  a  stone  tied  to 
the  end  of  a  string.  If  from  the  centre  of  gravity  of  the 
stone  (and  the  string)  to  the  point  of  suspension  is  39.1  inches, 
each  vibration  occupies  one  second.  For  any  other  length  L, 
each  vibration  occupies 

—  J—  seconds. 


386 


EXPLORATORY   SURVEYING. 


The  vibrations  should  be  kept  as  small  as  possible  so  as  to  re- 
duce the  resistance  of  the  atmosphere.  In  this  way  a  toler- 
ably long  interval  may  be  measured  with  a  fair  amount  of 
confidence.  The  best  way,  however,  is  to  compare  the  vibra- 
tions with  a  watch  subsequently. 


B.  BY  DIEECT  MEASUEEMENT  AND 
COMPASS  COUESES. 

1 75.  By  far  the  most  convenient  and  accurate  method  of 
obtaining  direct  measurement  on  exploratory  surveys  is  by 
means  of  an  odometer,  which  answers  the  same  purpose  as 
the  patent  log  at  sea,  only  more  efficiently;  but  unfortunately 
it  necessitates  the  use  of  some  wheeled  vehicle,  which  is  not 
always  a  convenient  appendage  to  an  exploring  outfit. 

Pedometers  answer  well  in  country  where  the  condition  of 
the  ground  is  comparatively  regular  and  walking  easy,  but 
where  the  surface  is  much  broken  they  are  worse  than  useless, 
being  misleading  as  well.  The  best  means  of  then  ascertaining 
the  distance  travelled  is  by  estimating  the  rate  of  progress  and 
keeping  track  of  the  time.  The  approximate  rate  may  always 
be  found  by  noting  the  time  occupied  in  covering,  say,  100 
yards  ;  then  if  t  —  the  time  occupied  in  seconds,  the  velocity  in 
miles  per  hour  equals 

200 , 
0  =  —-(nearly); 

so  that  we  have  the  following  values  of  v  for  various  values 
of  t: 


t 

V 

t 

V 

t 

V 

t 

V 

sees. 

m.  p.  h. 

sees. 

m.  p.  h. 

sees. 

m.  p.  h. 

sees. 

m.  p.  h. 

200 

1 

80 

2.5 

40 

5 

25 

8 

133 

1.5 

66 

3 

33 

6 

22 

9 

100 

2 

50 

4 

28 

• 

20 

10 

As  regards  keeping  the  courses  by  compass,  in  open  country, 
-it  is  best  to  establish  the  bearing  of  some  point  ahead  on  the 
probable  route  and  then  to  correct  it  by  estimation,  if,  when 
abreast  of  that  point,  it  should  be  found  to  be  considerably  to 


EXPLORATORY  SURVEYING.         187 

one  side  of  the  route  taken.  In  timber  country,  the  bearing  of 
the  sun  being  taken  from  time  to  time,  it  forms  a  highly  useful 
guide  when  no  distant  landmarks  are  visible.  At  night  the 
pole-star  forms  as  good  a  guide  as  could  be  wished  for. 


C.   BY  ASTKONOMICAL  OBSEKYATIONS. 

176.  Before  attempting  the  solution  of  astronomical  prob- 
lems in  connection  with  the  establishment  of  positions  on  the 
earth's  surface,  it  will  be  well  to  give  a  few  explanations  as 
briefly  as  possible  regarding  the  fundamental  principles  in- 
volved, and  definitions  of  the  terms  used. 

TIME. 

177.  Civil  or  Common  Time  is  really  what  is  termed  in 
astronomical  language  Mean  Solar  Time,  with  this  difference, 
that  a  civil  day  being  reckoned  from  midnight  to  midnight, 
the  corresponding  astronomical  day  is  reckoned  from  the  noon 
of  that  day  to  the  following  noon,  and  is  also  counted  con- 
tinuously up  to  24  hours.     Thus  4  A.M.  on  Jan.  10  would  be 
stated  in  mean  solar  time  as  161'  Om  Jan.  9.     Now  the  velocity 
with  which  the  earth  travels  round  the  sun  varies 'in  different 
parts  of  its  orbit.     Ov,  ing  to  this  cause  and  also  to  the  obliquity 
of  the  ecliptic  (see  Sec.  180)  the  sun's  apparent  motion  is  ir- 
regular.    Thus  we  find  that  the  sun  is  apparently  travelling 
faster  in  winter  than  its  average  rate,  and  in  summer  slower. 
It  is  simpler  to  consider  the  earth  as  stationary  and  the  celestial 
bodies  as  revolving  round  it.     In  speaking  of  the  velocity  of 
the  sun's  motion,   then,  it  is  its  motion  among  the  stars— or 
on  the  star  sphere— that  is  referred  to,  not  its  actual  motion 
in  the  sky;   the  average  rate  of  this  motion  is  about  59'  per 
day  and  in   a  direction  opposite  to  that  in  which  the  whole 
star  sphere  is  apparently  revolving,  so  that  the  motion  of  the 
sun  in  the  sky  is  really  slower  than  that  of  any  given  star,  the 
result  of  which  is  that  the  star  apparently  revolves  round  the 
earth  366  times  while  the  sun  only  makes  365  revolutions 
(nearly). 

Now,  owing  to  the  irregularity  in  the  sun's  motion,  it  is  more 
convenient  to  substitute  for  the  real  sun  a  fictitious  one,  termed 


188 


EXPLORATORY   SURVEYING. 


the  "  Mean  Sun,"  which  is  imagined  to  make  the  same  number 
of  revolutions  in  the  course  of  the  year  as  the  real  sun,  but 
always  to  maintain  the  same  rate  of  motion.  Thus  it  follows 
that  the  mean  sun  sometimes  crosses  the  meridian — i.e.,  is  due 
south — before,  and  sometimes  after,  the  real  or,  as  it  is  termed 
in  the  Nautical  Almanac,  the  apparent  sun. 

178.  The  interval  of  time  between  the  passage  of  these  two 
suns  across  the  meridian  is  called  the  Equation  of  Time, 
which  when  the  mean  sun  is  ahead  of  the  apparent  sun  is  con- 
sidered positive,  and  when  the  apparent  sun  is  ahead,  negative. 
Thus,  since  the  mean  sun  is  always  south  at  mean  noon,  by 
adding  or  subtracting  (as  the  case  may  be)  the  equation  of  time 
to  or  from  24  hours— subtracting  24  hours  if  necessary— we  ob- 
tain the  mean  solar  time  at  which  the  apparent  sun  is  on  the 
meridian,  i.e.,  apparent  noon.  Thus,  if  for  a  certain  day  the 
equation  of  time  is  given  as  -\-  12m  04%  the  apparent  sun  will 
be  on  the  meridian  12'"  04s  after  mean  noon,  or  at  O1'  12m  04s 
astronomical  mean  time.  Had  the  equation  been  negative,  ap- 
parent noon  would  have  occurred  at  23h  47m  56s  mean  astro- 
nomical time. 

Expressing  the  relative  positions  of  the  two  suns  in  the  form 
of  an  equation,  we  have 

Mean  Time  =  Apparent  Time  ±  Equation  of  Time. 

The  mean  time  of  that  sun  is  the  greater  whose  R.A.  is  the 
less.  (See  Sec.  180.) 


Day  of 
Month. 

Jan. 

Feb. 

March. 

April. 

May. 

June. 

1 
11 
21 

1 

11 
21 

_L  4m  0» 

4-  8  21 
+  11  41 

-j-  13™  54" 

4-14  29 
4-13  47 

4-  12«  28" 
4-10    06 
4-    7    12 

4-3*50" 
4-0    58 
-  1     25 

-  3™  03" 
-3    48 
-  3    37 

-  2m  24s 
-  0    36 
+1     31 

July. 

August. 

Sept. 

Oct. 

Nov. 

Dec. 

+  3m  36" 
4-5  15 
+  6  05 

+  6m  04" 
4-4  56 
4-2  53 

-  Om  13s 
-  3    85 
-  7    06 

-  10m27«  1-  16»19« 
-13    19-15    49 
-15    22-13    53 

-  I0»  39» 

-  G   as 

-  1     31 

The  above  values  of  the  Equation  of  Time  show  approxi- 
mately the  positions  of  the  two  suns  relatively  to  each  other 
throughout  the  year.  These  values  change  but  little  from 
year  to  year;  and  are  sufficiently  accurate  to  enable  an  engineer 


EXPLORATORY   SURVEYING.  189 

to  find  mean  time  to  a  few  seconds  whenever  he  may  not  have 
a  Nautical  Almanac  at  hand;  or  to  correct  the  reading  of  a 
sun-dial,  which  of  course  gives  apparent  solar  time,  in  order  to 
reduce  it  to  mean  time. 

1 71).  Now  the  interval  of  time  between  the  passage  of  a  star 
across  the  meridian  one  day  and  its  passage  on  the  following 
day  is  equal  to  one  Sidereal  day ;  and  since  the  sun  makes 
only  365.242  revolutions  to  366.242  of  the  stars,  we  have 
A  sidereal  day  =  23h  56m  48.09  mean  solar  time, 
or,         A  mean  day   =  241'  03m  5 6s. 55  sidereal  time; 
or,  in  other  words, 

To  convert  a  sidereal  interval  of  time  into  mean  solar  units, 
it  has  to  be  reduced  at  the  rate  of  9.830  seconds  per  hour; — 
while 

To  convert  a  mean  solar  interval  into  sidereal  units,  it  has 
to  be  increased  at  the  rate  of  9.856  seconds  per  hour. 

Sidereal  time  is  reckoned  from  the  "  vernal  equinox,"  or  the 
moment  at  which  the  sun  crosses  from  the  southern  to  the 
northern  hemisphere,  and  is  thus,  in  a  way,  altogether  inde- 
pendent of  mean  solar  time;  but  if  we  know  the  moment  at 
which  the  vernal  equinox  occurs  in  mean  time,  we  thus  have  a 
means  of  connecting  sidereal  with  mean  time.  But  instead  of 
having  to  start  our  calculations  from  the  vernal  equinox  each 
time,  the  sidereal  time  of  mean  noon  is  given  for  every  day  in 
the  year  in  the  Nautical  Almanac;  so  that 

To  convert  sidereal  time  into  mean  time,  we  have  this  rule: 
From  the  sidereal  time  given  (increased  if  necessary  by  24 
hours)  subtract  the  sidereal  time  at  the  preceding  noon,  and  then 
reduce  the  result  at  the  rate  of  9.830  seconds  per  hour; — and, 

To  convert  mean  time  into  sidereal  time:  Increase  the  mean 
time  at  the  rate  of  9.856  seconds  per  hour;  the  time  thus  ob- 
tained, added  to  the  sidereal  time  at  tfie  preceding  noon 
(subtracting  24  hours  if  necessary),  gives  the  corresponding 
sidereal  time. 

The  Conversion  of  the  Intervals  may  be  greatly  facilitated  by 
means  of  Table  XIX. 

DECLINATION  AND  RIGHT  ASCENSION. 

180.  These  are  terms  used  to  denote  the  positions  of  celestial 
bodies  in  the  star  sphere  relatively  to  the  equinoctial  (which  is 
really  its  "  equator")  and  a  plane  perpendicular  to  it  passing 


190         EXPLORATORY  SURVEYING. 

through  the  vernal  equinox;  in  the  same  way  as  terrestrial 
Latitudes  and  Longitudes  give  the  positions  of  places  on  the 
earth's  surface,  relatively  to  the  equator  and  the  meridian  of 
Greenwich. 

The  plane  of  the  earth's  equator  produced  to  the  star  sphere 
gives  what  is  called  the  Equinoctial;  and  the  Ecliptic,  which  is 
really  the  plane  occupied  by  the  earth's  orbit,  is  inclined  to 
the  equinoctial  at  an  angle  of  about  23°  27  (slightly  varying), 
which  is  termed  the  Obliquity  of  the  Ecliptic. 

Instead,  however,  of  expressing  the  Right  Ascension  of  bodies 
as  so  many  degrees  E.  or  W.  of  the  vernal  equinox,  it  is  more 
convenient  to  adopt  the  phraseology  of  sidereal  time  and  denote 
the  positions  of  bodies  according  to  the  interval  of  time  at 
which  they  cross  the  meridian  after  the  zero  of  sidereal  time, 
i.e.,  the  vernal  equinox.  Thus  it  follows  that  the  sidereal  time 
at  which  a  body  is  on  the  meridian  is  given  by  its  Right 
Ascension  (R.A.),  so  that  instead  of  speaking  of  the  "sidereal 
time  at  preceding  noon"  as  in  the  rules  given  in  Sec.  179,  we 
might  have  said  "the  R.A.  of  the  mean  sun  at  preceding 
noon,"  for  the  sidereal  time  at  noon  is  often  so  stated  in 
almanacs.  And  if  we  know  the  sidereal  time  at  mean  noon, 
say  at  Greenwich,  we  can,  by  adding  or  subtracting  the  equa- 
tion of  time  (as  the  case  may  be)  obtain  the  R.A.  of  the 
apparent  sun  at  mean  noon  at  Greenwich,  and  by  correcting 
the  sidereal  time  at  mean  noon  at  the  hourly  rate  of  +9.856 
seconds,  and  also  correcting  the  equation  of  time,  we  can  find 
the  sun's  R.A.  at  any  later  hour. 

The  Declination  of  a  body,  which  is  really  its  angular 
measure  on  the  star  sphere,  north  or  south  of  the  equinoctial,  is 
considered  positive  when  north,  and  negative  when  south. 

181.  But  so  far  we  have  assumed,  except  in  the  case  just 
mentioned  above,  that  it  has  been  unnecessary  to  correct  either 
the  equation  of  time,  R.A.  or  Dec.,  as  given  in  the  almanac; 
but  since  these  quantities  are  always  varying,  and  they  are 
only  given  for  a  certain  hour  at  a  certain  place,  when  required 
for  any  other  hour  the  values  as  given  in  the  tables  must  be 
corrected — usually  with  sufficient  accuracy  by  simple  inter- 
polation— to  reduce  them  to  the  time  for  which  they  may  be 
required.  And  since  every  15°  of  longitude  west  is  equivalent 
to  1  hour  later  and  15°  east  to  1  hour  earlier,  if  in  longitude 
90°  west  of  Greenwich  we  want  the  declination  of  the  sun  at 


EXPLORATORY    SURVEYING. 


191 


4  P.M.,  and  for  noon  on  that  day  it  was  given  in  the  almanac 
as  _j_  17°  40',  and  at  noon  on  the  following  day  as  -f  18°  00  , 
the  declination  at  4  P.M.  in  longitude  90°  west  (which  is 
equivalent  to  10  hours  later)  will  be  17°  48'. 3;  and  in  the  same 
way  the  R.A.  and  Equation  of  time  must  be  corrected. 

In  dealing  with  stars,  daily  and  hourly  corrections  are  un- 
necessary, since  their  Decs,  and  R.A.'s  change  but  little  in  the 
course  of  the  year  (see  Sec.  213);  but  in  dealing  with  the  moon, 
the  change  is  so  rapid  as  to  necessitate  a  more  accurate  inter- 
polation than  would  be  given  by  simple  proportion  as  above. 


HOUR  ANGLE,   ETC. 

182.  The  "hour-angle"  is  a  term  which  may  best  be  ex- 
plained by  means  of  Fig.  87. 


Fia.  87. 

Suppose  a  person  stationed  at  A,  on  the  earth's  surface,  ob- 
serves a  star  8  at  an  altitude  So.  above  the  horizon  ab.  Then 
if  P  is  the  celestial  pole  and  Z  ihe  zenith,  since  he  knows  the 
declination  of  the  star,  if  he  also  knows  his  latitude,  he  has  the 
three  sides  of  the  spherical  triangle  PZ8  given  by  the  comple- 
ments of  these  values;  and  this  triangle,  if  PZb  is  the  meridian 
of  A,  is  generally  known  as  the  astronomical  triangle,  and 
the  angle  ZP8  is  the  hour-angle,  which,  if  expressed  in  time, 
is  really  the  difference  in  R.A.  of  the  star  $and  of  a  point  on 
the  meridian  at  the  moment  of  the  observation;  or,  in  other 
words,  it  equals  the  difference  between  the  R.A.  of  the  star 
and  the  sidereal  time  at  the  moment.  Thus  if  the  hour-angle 


192         EXPLORATORY  SURVEYING. 

in  sidereal  time  =  H  and  the  local  sidereal  time  =  T,  we  have, 
to  convert  the  hour -angle  into  sidereal  local  time, 

T=  H-\-  R.A.  (-  24  hours  if  necessary); 
and  conversely, 

H—  r(-f  24  hours  if  necessary)  —  R.A., 

which  is  the  formula  for  obtaining  the  hour-angle  when  the 
body  observed  is  either  the  moon,  a  planet  or  star;  the  R  A. 
being  the  R.A.  of  the  body  observed  at  the  moment  of  obser- 
vation. In  the  case  of  the  sun,  in  order  to  convert  the  hour- 
angle  into  mean  local  time,  we  have  simply  to  reduce  it  to 
apparent  time  by  dividing  by  15  (as  given  below),  and  then 
apply  the  equation  of  time  (corrected  for  the  time  of  observa- 
tion) to  reduce  the  apparent  time  to  mean  time;  and  the  con- 
verse of  this — to  find  the  hour- angle  when  given  the  mean  local 
time— is  simply  a  reversal  of  the  process,  for  the  sun's  apparent 
time  is  its  hour-angle. 

The  value  h  of  the  hour-angle  in  angular  measure,  as  ob- 
tained for  instance  by  solving  the  astronomical  triangle,  must 
be  subtracted  from  360°  when  the  star  lies  in  east  in  order  to 
give  it  its  true  value.  Then  in  order  to  convert  h  into  Ht  since 
1  hour  is  equivalent  to  15°,  we  have 

7t  (in  degrees) 
// (in  hours)  =  — — ~5 -; 

and  this  equation  of  course  holds  good  if  for  the  words 
"hours"  and  "degrees"  we  substitute  on  both  sides  either  the 
word  "  minutes"  or  "seconds."  So  that,  for  instance,  if  we 
obtain  by  an  observation  of  a  star  in  the  east  a  value  for  the 
hour-angle—as  obtained  from  the  astronomical  triangle— of  40°, 
we  have  h  -  320°;  therefore  H  =  2P  20m. 

Table  XX  greatly  facilitates  the  conversion  of  H  into  h,  or 
vice  versa. 

183.  The  following  examples  serve  to  illustrate  what  has 
already  been  said. 

1.  At  what  hour  will  Arcturus  culminate  (i.e.,  be  on  the 
meridian]  on  Sept.  18,  1889,  at  Greenwich?  From  the  Nautical 
Almanac  we  find  that  the  sun's  mean  R.A.  at  mean  noon  at 
Greenwich  on  Sept.  18  —  llh  50nl  228.8,  and  also  that  the  R.A. 


EXPLORATORY  SURVEYING.          193 

of  Arcturus  will  then  =  14h  10m  37s. 8;  and  since  the  K.A.  of 
the  star  is  really  the  sidereal  time  at  which  it  culminates,  we 
have  merely  to  convert  its  R  A.  into  mean  time  according  to 
Sec.  182.  Thus  Arcturus  will  be  on  the  meridian  at  2h  20™  15s 
mean  astronomical  time,  i.e.,  at  21'  20™  15s  P.M. 

2.  Wliat  will  be  the  R.A.  of  the  apparent  sun  on  Nov.  15,  1889, 
in  longitude  90°  W.  at  4  P.M.?    Since  4  P.M.  in  90°  W.  occurs 
10  hours  after  mean  noon  at  Greenwich,  and  from  the  Nautical 
Almanac  we  find  the  Sun's  mean  R.A.  at  mean  noon  on  Nov. 
15  =  15h  39m  03s. 0.     Since  the  correction  for  10  hours  =  -f- 10 
X  9s. 856  —  lm  38".  5,  the  Sun's  mean  R.A.  corrected  to  date 
—  1511  4Qm  4is  5      Similarly  the  equation  of  time  corrected  to 
date  =  15m  08s. 3;  and  since  the  apparent  sun  is  then  ahead  of 
the  mean  sun,  the  R.A.  of  the  apparent  sun  for  the  date  re- 
quired =  15h  39™  40S.5  -  0"  15™  088.3  =  15h  24™  328.2. 

3.  Find  the  Sun's  decimation  at  8  A.M.  July  22,  1§89,  in 
longitude  30°  E.     Now  8  A.M.  at  30°  E.  occurs  6  hours  before 
mean  noon  at  Greenwich;  and  from  the  Nautical  Almanac  the 
declination  at  Greenwich  at  mean  noon  on  July  22d  =  -|-  20° 
12'  16",  which,  corrected  to  6  hours  earlier,  =  -f  20°  15'  15", 
which  is  the  declination  required. 

4.  Given  10'1  24'"  08s  as  the  local  astronomical  mean  time  on 
Feb.  1,  1889,  in  longitude  60°  W.  to  convert  it  into  local  sidereal 
time.     According  to  Sec.  179,  we  must  first  convert  this  time 
into  a  sidereal  interval  by  increasing  it  at  the  rate  of  9.856 
sees,  per  hour,  which  gives  10h  25m  50s. 5,  and  the  sidereal  time 
at  mean  noon  4  hours  later  than  Greenwich  mean  noon  =  20hl 
48m  11s. 2,  thus  the  local  sidereal  time  (deducting  24  hours) 
=  7h  14™  018.7. 

5.  Suppose  on  June  1,  1889,  we  observe  Castor  at  2h  30m  04* 
A.M.  local  time,  in  longitude  105°  W.  what  is  the  Itour-angle  in 
angular  measure? 

This  in  mean  astron.  time  equals,  May  31 14h  30m  048 

Increase  at  rate  of  98.856  per  hour 2™  228.9 

Sidereal  interval  in  sidereal  time 14h  32m  26».9 

Sidereal  time  at  mean  noon  in  105°  W.  May  31. .  4h  37™  50s.  7 

Sidereal  local  time  of  obs.  =  T. 19h  10m  178.6 

R.A.  of  Castor. .  .  7h  27™  328.7 


Hour-angle  J2"  (subtracting  24  hours) 2h  37™  508.3 

Therefore  Angular  equivalent  h  =. 39°  27'  35" 


194 


EXPLORATORY   SURVEYING. 


6.  Given  the  hour-angle  of  the  apparent  sun  in  the  east,  as 
obtained  from  the  astronomical  triangle,  as  14°  29'  10"  on  June 
14,  1889,  in  longitude  90°  E.,  find  tJie  mean  local  time.  Since 
the  observation  is  in  the  east,  h  =  345°  30'  50",  which  corre- 
sponds with  231'  02m  03s;  therefore  the  observation  occurred 
23h  02m  03s  apparent  time  after  apparent  noon  on  June  14;  and 
at  that  moment  the  mean  sun  was  ahead  of  the  apparent  sun 
by  Ora  10s,  therefore  the  mean  local  time  of  observation 
=  23h  02m  138  June  14. 


REFRACTION,    PARALLAX,    SEMI-DIAMETER,   AND 
DIP. 

184.  In  Sees.  51  and  165  we  have  already  considered  the 
effect  of  Refraction  when  dealing  with  objects  on  the  earth's 
surface.  The  same  uncertainty  exists  in  dealing  with  celestial 
objects  as  to  the  amount  of  the  correction  necessary  to  counter- 


Alt. 

Ref. 

!  Alt. 

Ref. 

Alt. 

Ref. 

Alt. 

Ref. 

Alt. 

Ref. 

Alt. 

Ref. 

000 

3300 

230 

1623 

630 

752 

1220 

416 

30 

1  38 

60 

033 

005 

3211 

235 

1604 

640 

741 

1240 

409 

31 

1  35 

61 

032 

010 

31  22 

240 

1545 

650 

731 

1300 

403 

32 

1  31 

62 

030 

015 

30  3(5 

245 

1527 

700 

721 

1320 

357 

33 

1  28 

63 

029 

020 

2950 

250 

1509 

710 

712 

1340 

351 

34 

124 

64 

028 

025 

2906 

255 

1452 

720 

703 

1400 

34(» 

35 

121 

65 

027 

030 

2823 

300 

1435 

730 

654 

1420 

340 

36 

1  18 

66 

025 

035 

2741 

305 

1419 

740 

646 

1440 

335 

37 

1  16 

67 

0-24 

040 

2700 

310 

1403 

750 

638 

1500 

330 

38 

1  13 

68 

023 

045 

2620 

315 

1348 

800 

630 

1530 

323 

39 

1  10 

69 

0  22 

050 

2542 

320 

1333 

810 

622 

1600 

317 

40 

108 

70 

021 

055 

2505 

325 

1319 

820 

615 

1630 

311 

41 

105 

71 

020 

1  00 

2429 

330 

13  05 

830 

608 

1700 

305 

42 

103 

72 

019 

105 

2354 

a-40 

1239 

840 

601 

1730 

259 

43 

101 

73 

017 

10 

2320 

350 

1214 

850 

5  55 

1800 

254 

44 

059 

74 

016 

15 

24  47 

400 

1150 

900 

549 

IH30 

249 

t:, 

057 

75 

015 

20 

22  15 

410 

1128 

910 

543 

1900 

2  II 

46 

055 

76 

014 

25 

21  44 

420 

1107 

920 

537 

1930 

2  tO 

47 

053 

77 

013 

30 

21  15 

430 

1047 

930 

531 

2000 

236 

48 

051 

78 

012 

35 

2046 

440 

1028 

940 

526 

2030 

232 

49 

050 

79 

Oil 

40 

2018 

450 

1010  , 

950 

520 

I  21  00 

228 

50 

048 

80 

010 

1  45 

1951 

500 

953 

1000 

515 

21  30 

224 

51 

046 

81 

009 

150 

1925 

510 

937 

1015 

508 

2200 

220 

52 

045 

82 

008 

1  55 

1859 

520 

921 

1030 

500 

2300 

214 

53 

043 

83 

007 

200 

1835 

530 

907 

1045 

454 

2400 

207 

54 

041 

84 

006 

205 

18  11 

540 

853 

1100 

447 

25  00 

202 

68 

040 

85 

005 

210 

1748 

550 

839 

11  15 

441 

2600 

1  56 

56 

038 

86 

004 

215 

1726 

600 

827 

11  30 

435 

2700 

1  51 

57 

037 

87 

003 

220 

1704 

610 

8  15 

11  45 

429 

2800 

147 

58 

036 

88 

002 

225 

1644 

620 

803  1 

1200 

423 

2900 

143 

5!) 

034 

89 

001 

I 

EXPLORATORY   SURVEYING. 


195 


act  the  refractory  power  of  the  air,  as  we  found  to  exist  when 
the  objects  observed  were  near  at  hand;  but  in  the  case  of 
Astronomical  Refraction  the  altitude  of  the  object  is  a  much 
more  important  factor  than  in  the  previous  case;  for  the  lower 
the  altitude  not  only  the  more  obliquely  do  the  "rays  pass 
through  the  successive  layers  of  air,  but  the  extent  of  atmos- 
phere which  they  have  to  traverse  is  greater  than  at  a  higher 
altitude.  The  preceding  table  of  Mean  Refractions,  calculated 
for  a  barometer  pressure  of  29.6  inches  and  a  temperature,  of 
50°  F.,  may  be  used  at  all  times  under  ordinary  circumstances, 
when  dealing  with  celestial  objects  whose  altitudes  exceed  30°. 
At  low  altitudes  the  corrections  given  in  the  table  should  be 
corrected  by  multiplying  them  by  the  factors  B  and  T,  which 
make  allowance  respectively  for  the  height  of  the  Barometer 
and  the  Temperature  of  the  air :  thus 

True  Refraction  =  Mean  Refraction  X  B  X  T. 

VALUES   OF  B. 


Bar.  In. 

28 

28.5 

29 

29.5 

30 

30.5 

31 

B 

0.946 

0.963 

0.980 

0.997 

1.014 

1.031 

1.047 

VALUES  OF  T. 


Temp. 

-  30°  F. 

-  10°  F. 

4-  10°  F. 

-f-  30°  F. 

+  50?  F. 

+  70°  F. 

+  90°F. 

T 

1.180 

1.130 

1.082 

1.038 

1.000 

0.960 

0.925 

The  correction  for  refraction  must  of  course  be  subtracted 
from  the  observed  altitude. 

185.  The  positions  of  all  celestial  bodies  as  given  in  the 
Nautical  Almanac  are  calculated  with  reference  to  the  Centre 
of  the  Earth  ;  thus  if,  as  in  Fig.  88,  an  observer  at  A  observes 
the  altitude  of  the  sun  8  to  be  the  angle  8AH,  in  order  to  re- 
duce this  angle  to  the  centre  of  the  earth,  i.e.,  to  the  angle 
SOh,  he  mnet  add  to  it  the  angle  A80,  which  is  termed  the 
Parallactic  angle. 

Now  if  8  were  just  on  the  horizon,  i.e.,  at  //,  then 

.  _  AO  _  Radius  of  Earth 

sin  AHO  -JJQ-  Distance  of  Sun' 


196 


EXPLORATORY   SURVEYING. 


where  AHO  is  termed  the  Horizontal  Parallax,  and  is  given 
in  the  Nautical  Almanac.     In  the  case  of  the  sun  it  varies 


FIG.  88. 

from  about  8". 7  to  9".0.     In  order  to  reduce  this  to  Parallax 
in  Altitude,  we  have  from  the  above  figure 

sin  A80  =  sin  AHO  sin  SAZ; 
therefore 

sin  (Par.  in  alt.)  =  sin  (Hor.  Par.)  cos  (alt.); 

or,  assuming  the  sines  of  small  angles  to  be  proportional  to  the 
angles  themselves, 

Par.  in  alt.  =  Hor.  Par.  X  cos  (alt.). 

Thus,  at  an  altitude  of  45°,  Parallax  in  altitude  —  6",  and 
at  60°  =  4". 

In  the  case  of  the  moon,  since  its  distance  from  the  earth 
compared  with  the  radius  of  the  latter  makes  it  important 
what  value  of  the  radius  is  used,  the  Hor.  Par.  is  given  in  the 
Nautical  Almanac  &$  Equatorial  horizontal  parallax,  meaning 
that  the  value  of  the  radius  used  is  that  at  the  Equator  ;  thus 
for  other  latitudes  the  correction  taken  from  the  following 
table  should  be  subtracted  from  it  before  applying  the  cor- 
rection for  altitude,  in  order  to  obtain  the  value  of  the  Hori- 
zontal parallax  suitable  for  the  latitude  in  question  : 


Eq.  Hor.  Par. 

LATITUDE. 

10° 

20° 

30° 

40° 

50° 

60° 

70° 

80° 

90° 

53' 
61' 

0".3 
0".4 

1".2 
1".4 

2".  7 
3".l 

4".  4 
5".l 

6".2 
7".2 

8".0 
9".2 

9".4 
10".8 

10".  3 
11".9 

10".  6 
12".  2 

EXPLORATORY   SURVEYING. 


197 


186.  Correcting  for  Semi-diameter.— In  taking  an  alti- 
tude of  the  sun,  the  upper  or  lower  "  limb"  is  generally  ob- 
served, and  the  altitude  so  obtained  corrected  by  the  subtrac- 
tion or  addition  of  the  semi-diameter — obtained  from  the 
Nautical  Almanac — to  reduce  it  to  the  sun's  centre.  In  observ- 
ing with  an  artificial  horizon,  the  application  of  the  correc- 
tion for  semi-diameter  can  be  avoided  by  bringing  the  reflec- 
tions to  coincide.  With  either  a  transit  or  sextant  a  good  way 
is  to  observe  one  limb  and  note  the  time,  and  immediately 
after  observe  the  other  limb  and  note  the  time ;  the  mean  alti- 
tude may  then  be  considered  to  give  the  altitude  of  the  sun's 
centre  at  the  mean  time. 

Similarly  in  observing  the  transit  of  the  sun  across  any 
vertical  plane  we  take  the  mean  time  of  the  passage  of  its 
east  and  west  limbs. 

In  observing  the  moon,  we  usually  can  only  observe  one 
limb  ;  and  in  this  case,  on  account  of  its  proximity  to  the  earth, 
it  is  necessary  to  apply  a  correction  to  the  semi-diameter  as 
given  in  the  Nautical  Almanac,  which  assumes  the  observer  to 
be  at  the  centre  of  the  earth,  in  order  to  allow  for  the  increase 
in  its  semi-diameter  on  account  of  his  being  nearer  to  it  than 
the  centre  of  the  earth.  This  is  termed  correcting  for  the 
Augmentation  of  the  Semi-diameter.  The  corrections  are 
given  in  the  following  table  : 


APPARENT  ALTITUDE. 

Semi-diam. 

10° 

20° 

30° 

40° 

50° 

60° 

70° 

80° 

90° 

14'  30" 

2".4 

4".7 

6".9 

8".8 

10".5 

11".8 

12".9 

13".5 

13".7 

17'   0" 

3".4 

6".  5 

9".5 

12".  1 

14".4 

16".  3 

17".  7 

18".6 

18".8 

In  finding  the  time  occupied  by  the  'semi-diameter  of  the 
sun  or  moon  in  crossing  the  meridian,  it  must  be  remembered 
that  it  is  only  when  the  declination  =  0°  that  (if  the  R.  A.  is  not 
changing)  the  semi-diameter  will  travel  across  the  plane  at 
the  rate  of  15°  to  one  sidereal  hour  (or  15°  2'  24"  to  one  mean 
hour).  At  any  other  declination  we  have,  as  the  rate  of  travel, 

15°  =  1  sid.  hour  X  cos  (dec.), 

on  just  the  same  principle  as  the  length  of  a  degree  of  longi- 
tude decreases  as  the  cosine  of  the  latitude.     In  the  same  way, 


198 


EXPLORATORY   SURVEYING. 


it  is  only  when  the  body  is  on  the  horizon  that  its  semi-diam- 
eter can  be  measured,  without  correction,  by  the  horizontal 
circle  of  a  transit,  for  as  the  altitude  of  the  body  increases,  so 
also  does  the  horizontal  circle  increase  its  reading  in  propor- 
tion to  the  secant  of  the  altitude. 

The  change  in  R.A.  during  the  passage  of  the  semi-diam- 
eter must  of  course  be  added  to  the  time  which  it  would  have 
occupied  had  its  R.A.  been  constant. 

187.  Dip. — This  is  a  correction  only  necessary  when  the  sea- 
level  is  taken  as  the  horizon,  and  is  practically  the  same  as  that 
given  in  Sec.  167.  It  is  to  be  subtracted  from  the  observed 
altitude.  The  following  are  its  approximate  values,  but  re- 
fraction enters  too  largely  into  the  question  to  enable  accuracy 
to  be  obtained  by  the  use  of  a  sea-horizon  : 


Height  above    1 
Sea-level  in  feet,  f 

5 

10 

20 

30 

40 

50 

60 

75 

Dip,  

2'  5" 

3'  0" 

4'  10" 

5'  10" 

6'  0" 

6'  40" 

7'  20" 

8'  10" 

Other  values  maybe  found  from  the  values  of  H,  calculated 
according  to  Sec.  167. 

188.  We  will  now  sum  up  the  corrections  (which  we  have 
already  considered)  necessary  to  apply  in  taking  ordinary  ob- 
servations. 

1.  Observation  for  Altitude. 

A.  Using  a  sea  horizon  or  level. 

If  a  Star.  Observed  Altitude  (—  Dip)  ±  Index-error  — 
Refraction  =  True  Altitude. 

If  the  Sun,  or  a  Planet.  Observed  Altitude  (—  Dip)  ± 
Index-error  —  Refraction  ±  Semi-diameter -[-(Hor. 
Parallax  X  cos  alt.)  =  True  Altitude. 

If  the  Moon.  Observed  Altitude  (  —  Dip)  ±  Index-error 
—  Refraction  -+-  (Hor.  Eq.  Parallax  corrected  for 
latitude  and  converted  into  Par.  in  alt.)  ±  Semi- 
diameter,  reduced  for  Augmentation  =  True  Alti- 
tude. 

B.  Using  an  artificial  horizon. 

In  this  case  the  double-altitude  as  read  on  the  arc -f- or  — 
the  Index-error  must  be  divided  by  2  in  order  to  obtain  the 
observed  altitude,  and  then  the  other  corrections — except  of 


EXPLORATORY  SURVEYING.  199 

course  for  Dip,  which  only  comes  in  when  using  a  sea-hori- 
zon—applied as  above.  If  the  two  reflections  are  brought  to 
coincide,  there  will  be  no  correction  needed  for  semi-diameter; 
but  a  more  perfect  observation  can  usually  be  obtained  by 
bringing  the  limb  of  one  reflection  in  contact  with  the  oppo- 
site limb  of  the  other,  in  which  case  the  semi- diameter  must 
be  corrected  for  as  above. 

"Index-error"  includes  errors  of  any  sort  in  connection  with 
the  instrument  for  which  allowance  must  be  made. 
2.  Observation  for  Azimuth. 

If  a  Star.     Observed  Azimuth  =  True  Azimuth, 
If  the  Sun  or  a  Planet.     Observed  Azimuth  ±  (Semi- 
diameter  X  sec  alt.)  =  True  Azimuth. 
If  the  Moon.     Observed  Azimuth  ±  Semi-diameter  (re- 
duced   for  Augmentation)   X    Sec.   alt.    =   True 
Azimuth. 

Having  now  considered  all  the  corrections  which  need  be 
applied  in  the  case  of  ordinary  field  observations  when  using 
either  a  sextant  or  small  portable  transit,  we  will  next  consider 
the  methods  by  which  the  latitude  and  longitude  of  a  place 
may  be  established  by  astronomical  observations, 

LATITUDE. 

189.  A.  By  a  Meridian  Altitude.— In  Fig.  88,  if  for  the 

moment  we  assume  the  observer  to  be  at  the  centre  of  the 
earth,  so  as  to  do  away  with  the  idea  of  parallax,  if  PSH  is 
the  meridian  and  S  the  Sun,  8E  represents  the  Sun's  Dec.  N. ; 
and  if  its  declination  did  not  change,  since  Sa  indicates  its 
path,  we  can  easily  see  that  its  altitude  would  be  greatest  when 
on  the  meridian.  But  since  its  declination  is  always  changing, 
the  Sun  attains  its  maximum  altitude  in  the  northern  hemi- 
sphere when  its  declination  is  changing  towards  the  north, after 
it  has  passed  the  meridian,  and  when  changing  towards  the 
south,  before  it  reaches  the  meridian.  The  difference  between 
its  meridian  altitude  and  its  maximum  altitude  does  not  ex- 
ceed at  any  season  1",  so  that  in  ordinary  work  the  maximum 
altitude  is  assumed  as  being  equal  to  the  meridian  altitude. 

In  taking  an  observation  of  the  moon  with  a  sextant  it  is 
necessary  to  allow  for  this,  especially  about  the  time  of  the 
equinoxes,  the  difference  between  its  meridian  and  maximum 
altitudes  sometimes  amounting  to  as  much  as  2'  15". 


200         EXPLORATORY  SURVEYING. 

When  a  transit  is  used  to  observe  the  meridian  altitude,  it  is 
usually  set  in  the  meridian,  so  that  no  correction  is  then  re- 
quired. 

For  the  amount  of  the  correction,  see  Note  G,  Appendix. 

Now  in  Fig.  88,  if  Oh  were  the  observer's  horizon,  the  alti- 
tude of  the  Sun  is  represented  by  the  angle  80k,  Zis  the  Zenith, 
and  the  latitude  of  the  place  of  observation  is  given  by  the 
angle  ZE.  Therefore  the  latitude  of  the  place  equals 

ES-\-SZ=  Dec.  N.  +  Zenith  distance. 

And  since  the  Zenith  distance  is  the  complement  of  the  alti- 
tude, we  are  thus  able,  by  means  merely  of  the  meridian  alti- 
tude, to  obtain  the  latitude;  and  this  applies  equally  well  to 
all  celestial  bodies,  so  that  in  the  northern  hemisphere,  if,  as 
Sin  Fig.  88,  the  Dec.  is  N.,  then 

Lat.  =  Dec.  N.  -f  Zenith  distance.  .  .  .  (a) 
If  declination  is  south,  as  8 ', 

Lat.  —  Zenith  distance  —  Dec.  8 (b) 

If  the  Star  is  above  the  Zenith,  as  S", 

Lat.  =  Dec.  N.  —  Zenith  distance (c) 

If  the  Star  is  below  the  pole,  as  8'", 

Lat.  —  Altitude  -|-  Co-declination (d) 

In  the  Southern  Hemisphere  the  same  formulae  apply,  bear- 
ing in  mind  that  what  is  South  in  the.  southern  hemisphere  is 
equivalent  to  what  is  North  in  the  northern. 

The  altitude  taken  "  below  the  pole"  is  of  course  the  mini- 
mum altitude.  The  altitudes  of  8"  and  S"'  are  observed  in 
the  north. 

Suppose,  for  instance,  we  observe  the  meridian  altitude  of 
Regulus  on  Mar.  17,  1889,  to  be  40°  16'  40". 

Now  the  declination  of  Regulus  at  that  date  =  12°  30'  30"  ; 
so  that  we  have 


EXPLORATORY  SURVEYING.         201 

Observed  altitude  of  Regulus 40°  16'  40" 

Correction  for  refraction —          1' 07" 


True  altitude  40°  15'  33" 

Therefore,  zenith  distance =  49°  44'  27" 

Declination  of  Regulus 12°  30'  30" 


Therefore,  Latitude  by  Eq.  (a) =  62°  14'  57"  N. 

Again,  suppose  on  Feb.  8,  1889,  in  longitude  105°  W.,  the 
meridian  altitude  of  the  sun's  upper  limb  is  observed  to  be 

48°  27'  20" 

Correction  for  refraction —  50" 

"  "  parallax r +  5" 

"  "  semi-diameter..  —        16' 15" 


True  altitude  of  sun's  centre 48°  10'  20" 

Therefore,  zenith  distance =  41°  49'  40" 

Now  the  sim's  declination  S.  at  Greenwich 

at  app.  noon  on  Feb.  8 =  14°  49'  30" 

Correction  for  7  hours  later. —          5'  36" 


Sun's  declination  at  date 14°  43'  54" 

Therefore,  Latitude  by  Eq.  (5) =  27°  05'  46"  K 

100.  It  is  always  preferable  to  use  a  star  instead  of  the  sun 
or  moon  for  a  meridian  altitude.  The  moon  should  only  be 
used  in  thick  weather,  when  the  stars  are  invisible.  In  select- 
ing a  star  for  the  observation,  the  altitude  should  not  be  less 
than  30°  if  possible,  on  account  of  refraction.  In  order  for  a 
star  to  appear  above  the  horizon  on  the  meridian,  the  sum  of 
the  declination  and  co-latitude  must  exceed  0°,  and  the  excess 
equals  the  true  altitude,  remembering  that  declination  north 
is  -f-  and  south  —  ;  this  gives  a  check  before  the  observation 
is  taken,  preventing  the  wrong  star  being  used.  For  stars  be- 
low the  pole  as  S'"  in  Fig.  88,  in  order  that  the  star  may  be 
visible  above  the  horizon  at  its- minimum  altitude  the  latitude 
must  exceed  the  co-declination,  the  excess  being  the  true 
altitude. 

When  using  a  transit,  we  may  proceed  in  two  ways : 

1.  By  observing  the  maximum  altitude  and  correcting  ac- 
cording to  Sec.  189,  and  Note  G,  Appendix. 

2.  By  setting  the  transit  in  the  meridian,  and  then  observ- 
ing the  altitude  of  the  passage. 


202 


EXPLORATORY   SURVEYING. 


The  meridian  may  best  be  obtained  by  an  Elongation  of 
Polaris  as  described  in  Sec.  57,  or  by  the  other  methods  de- 
scribed in  Sees.  57  and  202. 

In  taking  meridian  altitude  it  is  well  to  observe  a  star  in  the 
north  as  well  as  a  star  in  the  south  ;  the  mean  result  is  then 
tolerably  free  from  instrumental  errors. 

Polaris,  either  at  its  upper  or  lower  transit,  is  a  good  star 
to  use  on  account  of  its  slow  motion  admitting  of  several  alti- 
tudes being  taken. 

B.  By  Transits  across  the  Prime  Vertical. 

191.  This  is  the  most  accurate  method  of  obtaining  the 
latitude,  but  necessitates  the  use  of  a  transit. 


FIG.  89. 

In  Fig.  89  let  PZE  represent  the  meridian,  Z  the  zenith,  P 
the  celestial  pole,  and  8  the  body,  the  time  of  whose 
transit  across  the  prime  vertical— i.e.,  the  vertical  plane  ZO, 
lying  due  east  and  west — we  wish  to  observe,  in  order  by  it 
to  obtain  the  latitude.  Now  in  the  spherical  triangle  ZPS 
the  angle  at  P  =  the  hour-angle  h  (see  Sec.  182),  and  Z8  =  the 
co-alt,  of  the  body  when  on  the  prime  vertical,  ZP  the  co-lat- 
itude, and  PS  the  co-declination. 

Therefore,  since  Z  —  90°, 

tan  (lat.)  =  tan  (dec.)  X  sec  h. 

But  in  order  to  obtain  h,  we  must  know  the  exact  local  time 
of  the  observation,  which  may  be  obtained  according  to  Sees. 


EXPLORATORY   SURVEYING.  203 

195,  etc.  The  longitude  we  need  only  know  with  sufficient 
accuracy  to  admit  of  correcting  the  sidereal  time  at  mean 
noon,  i.e.,  for  ordinary  work,  to  about  20  miles. 

This  method  of  determining  the  latitude  of  a  place  admits 
of  high  precision,  since  an  error  of  1  second  in  the  local  time 
only  causes  an  error  of  about  If  seconds  in  latitude,  or  about 
170  feet. 

The  passage  of  the  star  across  the  prime  vertical  should  be 
observed  both  in  the  east  and  the  west  (or  else  another  star 
used),  and  the  mean  result  taken  to  eliminate  errors. 

The  altitude  of  a  body  when  on  the  prime  vertical  is  given 
by  the  equation 

sin  (alt.)  =  sin  (dec.)  cosec  (lat.); 

and  the  hour  at  which  the  observation  occurs  is  given  by  the 
equation 

sec  h  =  tan  (lat.)  cot  (dec.). 

If  the  transit  has  three  vertical  hairs,  which  it  should  at  least 
have  for  astronomical  work,  the  star  may  be  observed  at,  say, 
its  eastern  transit  on  the  north  side  of  the  prime  vertical  upon 
the  hair  which  is  to  the  left  of  the  collimation  centre  ;  then 
after  reversing  the  instrument,  the  star  may  be  observed  again 
on  the  same  hair.  If  the  telescope  is  left  in  the  last  position 
until  the  star  comes  to  its  western  transit,  it  is  observed  again 
on  the  same  hair  to  the  south  of  the  prime  vertical,  and  then 
reversing  the  telescope  the  star  again  crosses  the  same  hair  on 
the  north  side.  Thus  a  latitude  determination  is  arrived  at 
free  from  instrumental  errors  and  with  the.  errors  of  observa- 
tion greatly  reduced.  It  is  best  to  select  a  star  with  as  small 
a  declination  as  possible,  as  its  motion  in  azimuth  will  then 
be  more  rapid. 


C.  By  an  Altitude  out  of  the  Meridian. 

192.  It  often  happens  that  just  about  the  time  when  the  sun 
or  star  is  on  the  meridian  suitable  for  obtaining  the  latitude 
according  to  method  A,  it  becomes  obscured  by  passing 


204 


EXPLOKATORY   SURVEYING. 


clouds.     If,  however,  the  local  time  is  known  approximately, 

the  latitude  can  still    be    ob- 
tained in  the  following  way  : 

Suppose  in  Fig.  90  PZE  is 
the  meridian  and  S  a  star  which 
has  only  a  short  time  before 
crossed  the  meridian.  Then  in 
the  "astronomical  triangle" 
PZS,  if  we  know  ZS  =  co-alt., 
PS  =  co-dec,  and  the  hour-angle 
ZPS,  we  can  at  once,  by  solving  the  spherical  triangle,  find 
the  side  PZ  =  co-lat.  But  instead  of  using  the  common  for- 
mulae (as  given  in  Sec.  233),  the  following  will  be  found  sim- 
pler : 


FIG.  90. 


Make 


and 


tan  A  =  cos  ZPS  X  tan  PS, 


cos  B  =  cos  A  X  cos  ZS  X  sec  PS. 


Then,  if  the  six-o'clock  circle  and  the  prime  vertical  lie  on 
the  same  side  of  S,  as  will  always  be  the  case  when  S  is  near 
the  meridian, 

co-latitude  =  A  —  B ; 

but  if  S  lies  between  them,  we  have 

co-latitude  =  A  -f-  B. 

But  since  this  method  is  really  only  suitable  for  use  within  an 
hour  or  two  of  the  meridian  circle,  it  is  the  former  of  these 
two  equations  which  is  almost  exclusively  used. 

When  the  latitude  and  declination  are  of  contrary  signs, 
we  then  have  simply 

Lat.  =  (4 +5) -90°. 

To  use  this  method,  it  is  necessary  to  know  the  value  of  the 
hour- angle  with  tolerable  accuracy.  This  can  be  obtained  by 
one  of  the  methods  given  in  Sees.  195,  etc.;  or  in  the  case  of  a 
star  it  can  easily  be  obtained  by  observing  its  altitude  before 
reaching  the  meridian, — assuming  that  it  is  only  cloudy  about 
the  time  of  the  meridian  passage,— noting  the  time  by  an  or- 


EXPLOKATORY   SURVEYING.  205 

diuary  watch;  then  ou  the  other  side  of  the  meridian,  if  the 
moment  is  observed  at  which  it  again  reaches  the  same  alti- 
tude, half  the  in  terval  (converted  into  a  sidereal  interval)  — 
hour-angle  R(see  Sec.  182).  With  the  sun  this  is  only  appli- 
cable when  its  declination  is  changing  but  little,  or  when  near 
the  zenith. 

D.  By  double  Altitudes. 

193.  The  following  are  very  convenient  methods  of  ob^ 
taining  the  latitude  when  the  local  time  is  not  known. 

A.  By  two  altitudes  and  the  interval  of  time  between  them. — 
In  Fig.  91  let  Z  be  the  zenith,  P  the  celestial  pole,  /Sand  8' 
the  two  positions  of  the  star  at  the  moments  at  which  the  alti- 
tudes and  times  are  observed. 


FIG.  91. 

Then  the  interval  between  the  two  observations  in  sidereal 
time  =  the  hour-angle,  which  converted  into  angular  meas- 
ure =  8P8'.  Then  in  the  triangle  PS8't  SP  =  S'P  =  co- 
declination  ;  thus  we  can  find/S$'  and  PS' 8.  Then  in  the 
triangle  Z8'8,  since  we  have  the  three  sides  we  can  find  the 
angle  ZS'S,  which,  subtracted  from  PS '8,  gives  the  angle 
P8'Z.  Then  in  the  triangle  PS  'Z  we  have  8  'P,  8  'Z,  and  the 
angle  P8  'Z,  from  which  we  can  find  PZ  =  co-latitude. 

A  good  common  watch  is  all  that  is  required  to  observe  the 
intervals. 

But  instead  of  taking  two  altitudes  of  the  same  star,  it  is 
better  to  observe — 

B.  By  simultaneous  altitudes  of  different  stars.— The  hour- 
angle  is  given  by  the  difference  in  R.A.  of  the  two  stars,  and 
the  rest  of  the  working  is  the  same  as  above.  When,  how- 
ever, there  is  but  one  observer,  so  that  the  altitudes  must  be 
taken  in  succession,  he  must  proceed  thus:  The  altitude  of 
one  star  must  be  taken,  and  the  time  noted  by  the  watch;  the 


206  EXPLORATORY   SURVEYING. 

altitude  of  the  other  star  must  then  be  taken,  and  the  time 
again  noted.  After  a  short  interval  the  altitude  of  the  second 
star  must  again  be  taken,  and  the  time  noted,  fie  thus  finds 
the  motion  in  altitude  of  the  second  star  in  a  given  time,  from 
which,  by  proportion,  he  can  find  what  its  altitude  was  when 
the  first  star  was  observed. 

In  both  A  and  B  the  altitudes  as  observed  must  of  course 
be  reduced  to  the  true  altitudes  in  order  to  obtain  SZ  and 
B\Z. 

194.  On  the  last  page  of  the  Nautical  Almanac  for  each  year 
is  given  a  Table  for  computing  the  latitude  from  an  observed 
Altitude  of  Polaris  at  any  time,  the  hour-angle  being  approx- 
imately known;  and  as  full  instructions  accompany  the  table, 
these  need  not  be  repeated  here.     The  local  time  being  known, 
the  hour-angle  H  is  of  course  -obtained  as  in  Example  5,  Sec. 
183. 

LONGITUDE. 

195.  The  simplest  way  of  obtaining  the  longitude  of  a 
place  is  to  find  its  correct  local  time,  and  compare  it  with  a 
chronometer  which  gives  Greenwich  time  ;  the  difference  be- 
tween the  two  times  equals  the  difference   of  longitude :  so 
that  if  we  have  a  chronometer  at  hand  keeping  Greenwich 
time,  obtaining  the  longitude  is  simply  a  matter  of  obtaining 
the  local  time. 

A.  To  obtain  Local  Time  by  an  altitude  of  a  star. 

If  it  were  not  for  the  slowness  of  the  motion  of  a  star  when 
near  the  meridian,  a  convenient  method  of  obtaining  the  local 
time  would  be  to  reduce  its  R.A.  to  mean  time  at  the  mo- 
ment of  its  maximum  altitude,  which  would  then  be  the  mean 
local  time  of  its  transit.  But  in  order  to  obtain  a  well-defined 
moment  of  observation,  it  is  necessary  for  the  motion  in  alti- 
tude to  be  as  rapid  as  possible,  and  for  this  reason  a  star 
snould  be  selected  as  near  the  prime  vertical  as  possible.  Sup- 
pose at  a  certain  moment  by  the  chronometer  we  observe  the  al- 
titude of  a  star  8  (see  Fig.  90);  then  if  the  latitude  is  known, 
in  the  triangle  PZ8,  since  PZ  =  co-hit.  =  I,  PS  =  co-dec.  = 
d,  and  SZ  =  co-alt.  =  a,  we  have,  by  spherical  trigonometry, 


7i       A  /sin  s  sin  (s  —  a) 

COS  ^  =  A/ .  \ — —^ 

2       V        sin  d  sin  I 


EXPLORATORY   SURVEYING. 


207 


where  s  = 


and  h  =  the  hour-angle  ZPS;  if  the  dec- 


lination and  the  latitude  are  of  opposite  signs,  d  =  dec.  +  90°. 

Now  the  nearer  8  is  to  the  prime  vertical,  the  less  is  an  ac- 
curate knowledge  of  the  latitude  essential,  and  the  less  does 
an  error  in  altitude  affect  the  result.  Thus  the  body  should 
be  observed  as  nearly  east  or  west  as  possible,  and  certainly 
not  within  an  hour  or  two  of  its  transit. 

The  following  table  shows  the  errors  in  longitude  in  min- 
utes of  arc  involved  by  an  error  of  1  minute  in  latitude,  when 
S  is  observed  at  different  bearings  in  different  latitudes. 


LATITUDE. 

Bearing. 

10° 

20° 

30° 

40° 

50° 

60° 

70° 

10° 

5'.67 

5'.  76 

6'.55 

7'.40 

8'.  82 

11'.33 

6'.  03 

20° 

2'.75 

2'.79 

3M7 

3'.59 

4'.27 

5'.49 

2'.  92 

40° 

1'.19 

I'JU 

1'.38 

1'.55 

1'.85 

2'.38 

1'.27 

60° 

0'.58 

O'.oQ 

0'.67 

0'.75 

0'.90 

1'.15 

0'.62 

80° 

OM8 

O'.IS 

0'.20 

0'.23 

0'.27 

0'.35 

0'.19 

Thus  in  latitude  30°  if  the  bearing  of  a  star  when  observed 
is  80°  an  error  in  latitude  of  5  miles  would  only  cause  an  error 
of  about  half  a  mile  in  longitude. 

An  error  in  the  altitude  is  of  much  more  importance,  as 
the  following  table,  giving  the  errors  in  longitude  in  minutes 
caused  by  an  error  of  one  minute  in  altitude,  -shows: 


LATITUDE. 

10° 

20° 

30° 

40° 

50° 

60° 

70° 

10° 

5'.91 

6'.25 

6'.  65 

7'.  50 

8'.96 

12'.  17 

16'.  87 

30° 

2'.  03 

2'.  17 

2'  30 

2'.  64 

»*.  14 

3'.  98 

5'.84 

50° 

I'M 

I  '.39 

1'.51 

1'.71 

2'.03 

2'.63 

3'.78 

90° 

I'.Ol 

I'.oe 

I'.IS 

1'.31 

1'.55 

2'.00 

2'.90 

Since  the  accuracy  of  the  altitude  is  of  great  importance,  it 
is  well  to  take  several  sights,  say  3  or  5,  within  a  minute  or  so 
of  each  other,  and  note  the  corresponding  chronometer  read- 
ings ;  the  mean  altitude  may  then  be  considered  to  correspond 
with  the  mean  time.  If  the  local  time  which  was  used  in 
order  to  correct  the  sidereal  time  at  noon  for  the  assumed 


208  EXPLORATORY    SURVEYING. 

longitude  is  found  to  have  been  appreciably  in  error,  allow- 
ance must  be  made  for  this. 

In  observing  altitude  for  time,  if  great  accuracy  is  desirable, 
it  is  well  to  observe  both  in  the  east  and  the  west ;  the  mean 
result  of  the  two  sets  is  thus  practically  free  from  instrumen- 
tal errors.  This  method  of  course  applies  equally  well  to  the 
sun  as  to  a  star;  and  since  the  co-declination  is  always  a  large 
arc,  whatever  error  there  may  be  in  it,  there  will  only  be  half 
that  error  in  the  half  sum;  and  since  the  errors  in  these  alti- 
tudes oppose  one  another,  an  error  in  the  co-declination  such 
as  might  arise  from  an  error  of  two  or  three  degrees  in  the 
longitude  assumed  to  correct  the  sun's  declination  will  not 
seriously  affect  the  result. 

B.  To  obtain  local  time  by  equal  altitudes  of  a  star. 

196.  All  that  we  have  to  do  in  this  case  is  to  observe  the 
altitude  of  a  star  in  the  east  and  note  the  time,  then  note  the 
time  when  in  the  west  it  again  descends  to  the  same  altitude. 
Half  the  interval  between  the  two  observations  is  the  "mid- 
dle-time," which  corresponds  with  the  local  sidereal  time 
given  by  the  star's  R.A.    Thus  we  have  simply  to  convert  the 
star's  R.A.  into  mean  local  time  and  compare  it  with  the 
middle-time  by  the  watch  to  obtain  the  watch-error. 

By  taking  a  set  in  the  east  and  a  set  in  the  west,  since  index 
or  instrumental  errors  do  not  enter  into  the  question  at  all, 
the  mean  altitude  for  the  mean  time  should  give  a  really 
good  result.  There  is  no  necessity  to  apply  a  correction  for 
refraction,  unless  the  barometric  pressure  or  temperature  has 
changed  considerably  between  the  observations. 

C.  To  obtain  local  time  with  a  transit. 

197.  The  best  way  to  proceed  with  a  transit  is  to  set  it  in 
the  meridian  and  observe  the  time  of  transit  of  the  sun  or  one 
or  more  stars;  the  correct  local  time  is  then  found  by  merely 
converting  the  R.A.  of  the  body  at  the  time  of  its  transit  into 
mean  time. 

198.  But  so  far  in  obtaining  the  longitude  we  have  assumed 
that  we  have  had  at  hand  a  chronometer  rated  to  Greenwich 
time.     But  since  little  reliance  can  be  placed  on  chronometers 


EXPLORATORY   SURVEYING  209 

when  travelling  across  country,  one  of  the  following  methods 
should  be  adopted  as  a  check  on  the  chronometer  from  time 
to  time. 

TO  OBTAIN  THE  LONGITUDE  BY  LUNAR 
CULMINATIONS. 

The  principle  on  which  this  method  of  obtaining  Green- 
wich time  is  based  is  as  follows: 

In  the  Nautical  Almanac  the  moon's  R.A.  is  given  for  every 
hour  during  the  year  at  Greenwich.  If  then  in  any  other 
longitude  we  find  the  moon's  R.A.  at  a  certain  moment  that 
moment  will  correspond  with  the  time  at  Greenwich  at  which 
the  moon  would  have  the  same  R.A.  as  that  which  we  ob- 
served. Thus,  if  the  moon's  R.A.  in  the  Almanac  at  6  p  M 
were  given  as  8",  if  in  a  certain  longitude  we  find  at  exactly 
0  P.M.  local  time  the  moon's  R.A.  to  be  8h,  we  know  we  are 
in  a  longitude  4  hours  ahead  of  Greenwich,  i.e.,  60°  E  To 
obtain  the  R.A.  of  any  body  by  observation,  we  have  only 
to  find  the  mean  local  time  of  its  transit  across  the  meridian 
and  convert  it  into  sidereal  time,  which  is  the  R.A.  required 
Thus  we  proceed  as  follows: 

Find  the  correct  local  time  by  the  watch.  Set  the  transit  in 
the  meridian.  Observe  the  moment  of  transit  of  the  moon's 
bright  limb.  Again  find  the  correct  local  time  by  the  watch 
The  moon's  semi-diameter,  which  is  given  for  every  12  hours 
m  the  Almanac,  must  then  be  found  and  divided  by  15  to  re- 
duce it  to  equivalent  time,  which  would  then  be  the  sidereal 
time  occupied  by  its  passage  if  its  declination  =  0°  and  its 
R.A.  were  unchanging.  But  since  its  R.A.  is  always  increas- 
ing, the  passage  of  the  semi-diameter  will  occupy  a  time 
longer  than  this  by  an  amount  which  ma*  be  obtained  from 
the  Almanac  by  simple  proportion,  by  seeing  what  the  increase 
m  R.A.  is  at  the  assumed  Greenwich  time  of  the  observa 
tion;  the  total  time  of  the  passage  so  obtained  multiplied  by 
the  secant  of  the  declination  (see  Sec.  186)  then  gives  the 
time  actually  occupied  in  the  passage;  and  this  added  to  or 
deducted  from,  the  observed  time  of  transit  of  the  limb  gives 
the  time  of  transit  of  the  moon's  centre,  which,  converted  into 
sidereal  time,  gives  the  moon's  R.A.  at  the  moment  of  obser- 
vation. 


210  EXPLORATORY   SURVEYING. 

It  is  well  to  take  a  set  of  altitudes  for  time  before  and  after 
the  moon's  passage;  and  the  instrument,  if  possible,  should 
not  have  less  than  3  vertical  hairs,  the  passage  across  each  of 
which  may  be  observed  and  reduced  to  the  centre  hair, 

Every  possible  precaution  should  be  taken  in  this  obser- 
vation, for  the  error  of  a  second  of  time  in  observing  the 
moon's  limb,  compared  with  the  corrected  watch  time, 
— i.e.,  an  error  of  1  second  in  R  A., — may  easily  cause  an 
error  in  longitude  of  5  miles.  Thus  by  a  single  observation 
with  a  small  transit  we  cannot  depend  on  our  longitude  to 
within  about  10  miles.  But  if  the  observer  is  stationed 
for  3  or  4  days  at  any  one  place,  by  taking  the  mean  result  of 
3  or  4  observations  he  should  be  able  to  obtain  the  longitude 
with  a  probable  error,  say,  not  exceeding  4  or  5  miles,  corre- 
sponding with  an  error  in  Greenwich  time  (in  ordinary  lati- 
tudes) of  from  20  to  30  seconds. 

Having  now  obtained  the  moon's  R.A.,  the  next  thing  to  do 
is  to  find  the  hour  at  Greenwich  with  which  it  corresponds. 

Since  the  moon's  change  in  R.  A.  is  usually  rapid,  and  great 
accuracy  is  necessary,  the  ordinary  method  of  simple  interpo- 
lation will  not  apply  here.  The  following  formula  may 
therefore  be  used  instead: 

60  (A  -  a) 


T-t  = 


-       d(T-l\> 
h  2  \  3600  ) 


where  T  —  the  hour  required; 

t  =  the  hour  for  which  R.A.  is  given  in  the  Almanac, 

previous  to  T; 

A  =  R.A.  corresponding  with  T; 
a  =  R.A.  corresponding  with  tf; 
D  —  Increase  in  R.A.  in  1  mean  minute  at  time  t\ 
d  —  Increase  in  D  in  1  mean  hour  at  time  t. 
If  D  is  decreasing,  d  is  of  course  negative.     In  the  term  in- 
volving the  unknown  value  (T  —  t),  the  probable  value  must 
be  used,  which  is  correct  enough.     We  thus  have  the  value 
of  the  Greenwich  time  corresponding  with  the  observed  local 
time  of  the  transit  of   the  moon's  centre,   the  difference  of 
which,  divided  by  15,  gives  the  difference  of  longitude. 

199.  TO  OBTAIN  THE  LONGITUDE  BY  LUNAR  DIS- 
TANCES.—This  method  is  similar  in  principle  to  the  preced- 


EXPLORATORY   SURVEYING.  211 

ing  one,  the  difference  being  that  here  it  is  the  distance  from 
the  moon  to  some  star  which  is  observed  instead  of  its  R.  A. 
The  present  case,  since  it  does  not  involve  the  use  of  a  transit 
and  admits  of  several  observations  being  taken  on  one  night, 
is  more  suitable  for  exploratory  work,  and  is  the  method  alto- 
gether used  for  checking  the  chronometers  at  sea.  The  dis- 
tances between  the  moon's  centre  and  certain  stars  of  the  first 
and  second  magnitude  are  given  in  the  Nautical  Almanac  for 
every  three  hours  at  Greenwich,  so  that  it  is  simply  a  case  of 
measuring  the  distance  from  the  moon's  limb  to  a  star,  and 
correcting  for  refraction,  semi-diameter,  etc.,  noting  the  local 
time  of  the  observation,  and  then  finding  from  the  Almanac 
what  hour  at  Greenwich  corresponds  with  the  corrected  dis- 
tance. 

In  Fig.  92  let  M'  and  8'  be  the  positions  of  the  moon 
and  star  at  the  moment  of  observa- 
tion, and  Z  the  zenith;  then  M'S' , 
corrected  for  semi-diameter,  equals 
the  apparent  Lunar  distance,  and  M  'Z 
and  8'Z  the  co-altitudes.  The  true 
positions  will  differ  from  these  by 
the  differences  in  altitude  MM'  and 
88':  the  moon,  on  account  of  the 
correction  for  parallax  exceeding 
that  for  refraction,  will  be  elevated 
above  its  apparent  position ;  whilst  the  star,  on  account-  of  re- 
fraction only,  will  be  depressed  below  its  observed  position. 

Now,  if  the  apparent  altitudes  are  observed  at  the  time  of 
observing  the  lunar  distance  8'M',  we  have  the  three  sides  of 
the  triangle  8'ZM' ,  so  that  the  angle  at  Zm&y  be  found  trigo- 
nometrically.  Then  the  two  sides  8'Z  and  M' Z,  being  cor- 
rected for  refraction  and  parallax,  give  the  sides  of  the  cor- 
rected triangle  8ZM;  and  since  we  thus  have  two  sides  and 
the  included  angle  Z,  we  can  calculate  the  true  lunar  distance 
SM.  This  operation  is  termed  "  Clearing:  the  lunar  dis- 
tance." 

The  following  formula,  by  Borda,  is  probably  the  most  con- 
venient to  use  for  effecting  this: 

D  H+H' 

sin  —  =  cos  — cos  0. 

A  % 


EXPLORATORY   SURVEYING. 
where 


•  2  ri  _  cos  s  cos  (8  ~  ^  cos  ^7cos  H' 
cos  A  cos  h'  cos5  —  ^t  - 


where  s  = 


and  A  =  app.  alt.  of  moon's  centre,  h'  =  app.  alt.  of  star; 
H=  true  alt.  of  moon's  centre,  H'  =  true  alt,  of  star; 
d  =  app.  distance  S'M',  D  —  true  distance  8M. 

An  error  of  a  minute  or  two  in  the  altitude  makes  no  appreci- 
able difference  in  the  distance. 

The  vernier  should  be  set  to  a  division  easily  read  off,  and  at 
the  moment  when  the  distance  agrees  with  this  reading  the  ob- 
server should  call  "  stop,"  at  which  signal  the  assistant  should 
note  the  time  by  the  watch,  and  at  the  same  instant,  if  possible, 
the  altitudes  may  be  observed  by  two  assistants.  But  usually 
one  observer  has  to  do  the  whole  work  with  the  sextant,  in 
which  case  he  will  have  to  observe  the  altitudes  of  the  moon 
and  star,  both  before  and  after  the  observation,  and  note  the 
times,  and  then  deduce  the  altitudes  at  the  time  of  measuring 
the  distance,  by  proportion. 

But  a  better  way  is  to  spend  the  time  otherwise  occupied  in 
observing  altitudes,  in  obtaining  a  large  number  of  lunar  dis- 
tances and  then  to  compute  th£  altitudes  as  follows: 

Since  we  know  the  time  of  each  observation,  we  can  obtain 
the  hour-angle  at  that  moment,  which,  in  either  the  case  of  the 
moon  or  a  star,  is  merely  the  difference  in  R.A.  of  the  body 
and  the  sidereal  time  at  the  moment  -1-  24  hours  if  necessary, 
the  R.A.  in  the  case  of  the  moon  being  corrected  for  the  time 
of  observation  by  assuming  a  probable  value  for  the  longitude. 
Then  if  L  —  latitude  and  d  =  co-declination, 

sin  L  sin  (E+d) 
sin  (alt.)  =        -^— ^ , 

where 

cot  E  =  cot  L  cos  h, 

and  h  =  the  hour-angle.    If  h  exceeds  90°  cos  h  is  negative, 
which  will  make  cot  2£also  negative;  so  that  to  avoid  the  use 


EXPLORATORY   SURVEYING.  213 

of  supplements,  it  is  simpler  to  say 

sin  L  sin  (E  -  d) 


sin  (alt.)  = 


sin  E 


These  are  of  course  the  true  altitudes. 

In  selecting  stars  from  which  to  measure  the  distance,  it 
should  be  remembered  that  the  mean  of  two  distances,  one 
measured  to  a  star  on  the  right  and  the  other  on  the  left,  will 
be  practically  free  from  instrumental  errors;  so  that  this  plan 
of  observing  should  always  be  adopted  when  possible.  It  is 
well,  too,  to  select  stars  the  distances  between  which  and  the 
moon  are  varying  most  rapidly, — for  there  is  a  considerable 
difference  sometimes  between  the  rates, — and  yet  at  the  same 
time  the  altitudes  should  not  be  less  than,  say,  10°. 

A  complete  lunar  observation  should  consist  of  6  "  sets," 
each  set  including  3  simple  distances;  3  of  these  sets  should  be 
taken  to  the  left  of  the  moon  and  3  to  the  right;  also  two  ob- 
servations for  latitude,  one  in  the  north  and  one  in  the  south, 
to  eliminate  instrumental  errors;  and  two  sets  of  observations 
for  time,  one  to  a  star  in  the  east  and  another  in  the  west,  one 
before  and  the  other  after  the  measuring  of  the  distances. 

Having  thus  obtained  the  mean  lunar  distance  for  the 
mean  local  time,  the  corresponding  Greenwich  time  may  best 
be  deduced  according  to  the  instructions  and  data  given  in  the 
Nautical  Almanac  with  sufficient  clearness  to  render  any 
further  explanation  superfluous,  as  that  work  must  of  necessity 
be  an  accompaniment  to  the  observations.  Since,  however,  the 
Nautical  Almanac  assumes  that  the  computer  has  at  hand  a  table 
of  Ternary  Proportional  Logarithms,  such  as  is  given  in 
Chambers'  Mathematical  Tables  or  Bowditch's  Navigator,  it 
will  be  well  to  see  how  these  may  be  calculated,  in  the  event  of 
such  not  being  the  case. 

A  Proportional  Logarithm  for  any  portion  of  a  certain 
period  is  merely  the  difference  of  the  logarithms  of  the  period 
and  of  the  portion.  Thus,  taking  the  period  as  3  hours,  since 
lunar  distances  are  given  in  the  Almanac  at  intervals  of  every  3 
hours,  or  10,800  seconds,  the  logarithm  for  it  =  4.0334;  then 
since  the  logarithm  for  1  hour  (=  3600  seconds)  =  3.5563,  the 
proportional  logarithm  for  1  hour  =  0.4771.  The  explorer, 
however,  should  provide  himself  with  some  portable  form  of 


214  EXPLORATORY   SURVEYING. 

logarithmic  tables  if  likely  to  have  much  of  this  sort  of  work 
to  do. 

200.  Another  method  of  obtaining  Greenwich  time  is  by 
observing  with  a  powerful  telescope  the  local   time  of  the 
Eclipses  of  Jupiter's  Satellites.    But  this  method,  for  a 
variety  of  reasons,  is  considerably  less  reliable  than  those  given 
above.     The  Nautical  Almanac  gives  instructions  and  data  as 
to  the  manner  of  obtaining  Greenwich  time  by  this  method. 

TO  TEST  THE  CHRONOMETER  RATE. 

201.  Whenever  a  halt  is  made  for  over  24  hours,  it  is  a  very 
simple  matter  to  check  the  rate  of  the  chronometer.     With  a 
transit  this  can  best  be  done  by  setting  it  in  a  vertical  plane 
lying  fairly  north  and  south,  and  noting  the  moments  of  the 
passages  of  3  or  4  stars.     The  interval  of  time  before  the 
respective  passage  of  each  on  the  following  evening  =  23h  56m 
04s. 9.     With  a  sextant  this  may  best  be  done  by  observing  the 
altitudes  of  3  or  4  stars  lying  fairly  east  or  west— their  motion 
being  greater  in  altitude  when  near  the  prime  vertical — and 
noting  the  chronometer  times;  after  the  lapse  of  the  above  in- 
terval, each  will  again  be  at  the  same  altitude  on  the  following 
night. 

TO  SET  THE  TRANSIT  IN  THE  MERIDIAN. 

202.  Three  methods  of  obtaining  a  north  and  south  line 
have  already  been  given  in  Sec.  57;  the  method  by  Maximum 
Elongations  of  Polaris   is  the  best,  for  it  admits    of  plenty 
of  time  to  reverse  the  instrument  and  establish  a  true  north 
and  south  line.     When  Polaris   is   not   convenient   for  this 
purpose,  any  other  star  (which   has  an  elongation)   may  be 
used  as  shown  in  Note  D,  Appendix.     In  the  same  way,  if 
neither  Alioth  nor  y  Cassiopeia  is  convenient  for  observation, 
other  stars  may  be  used  as  shown  in  Note  E,  Appendix.   When, 
however,   neither  of  these  methods  is  exactly   suitable,  the 
azimuth  of  Polaris  out  of  the  meridian  may  be  found  at  any 
moment  by  solving  the  astronomical   triangle  PZ8  in  Fig.  87, 
and  thus  obtaining  the  angle  at  Z,  which  is  the  azimuth. 

To  do  this  we  have  given  the  declination,  and  we  must  also 
have  two  of  the  following  three :  latitude,  altitude,  and  hour- 
angle.  Since  the  latitude  is  most  easily  obtained,  and  the 


EXPLORATORY   SURVEYING.  215 

altitude  gives  the  best  result  if  near  the  elongations,  these  two 
should  then  be  used.     If,  however,  the  star  is  near  the  meridian, 
the  latitude  and  the  hour-angle  should  be  employed. 
In  the  former  case  we  have 


Z          /  sin  s  sin  (s  —  d) 
DS  2  ~  V  ~      sin  a~smT~ 

a,  d,  and  I  being  the  complement  of  the  altitude,  declination 
and  latitude  respectively,  and  s  the  half  sum  of  a,  d,  and  I. 
In  the  latter  case  we  have 

cos  a  =  cos  d  cos  I  -f-  sin  d  sin  I  cos  #, 
from  which  we  obtain 

sin  Z  =  sin  7i  sin  d  cosec  a. 

h  =  hour-angle.     (See  Sec.  182.) 

When  the  latitude  and  declination  are  of  opposite  signs, 
d  =  dec.  -f  90°. 

203.  In  observing  the  altitude  of  the  moon  for  time  or 
latitude,  as  is  often  practicable  in  thick  weather  when  the  stars 
are  invisible,  and  more  accurate  interpolation  of  its  declination 
is  necessary  than  is  obtained  by  simple  proportion,  the  method 
usually  adopted  for  this  purpose  is  that  known  as  INTER- 
POLATION BY  SUCCESSIVE  DIFFERENCES.  The 
interpolation  formula  is 

ndl        n(n  -  1)  n(n  -  1)  (n  -  2) 

=  F+  -T  +    1x2    <**  +     1x2x^3—  d3  +'  etc" 

For  example,  suppose  we  wish  to  find  the  moon's  declina- 
tion at  Greenwich  at  2h  15™  on  Nov.  15,  1889. 

From  the  Nautical  Almanac  we  find  Uie  declination  given 
for  every  hour.  We  select  the  declination  at  the  hour  before 
the  one  for  which  we  wish  to  interpolate  (=  F),  and  put  it  in 
the  first  column  as  below  ;  beneath  it  we  put  in  order  the  decli- 
nations for,  say,  3  or  4  following  hours,  as  given  in  the  Almanac. 
In  the  second  column  we  put  down  the  first  differences  of 
these  (di)  obtained  by  subtracting  downwards  and  prefixing  the 
proper  algebraic  sign.  In  the  third  column  we  place  the 
second  difference  (d^)  (i.e.,  the  differences  of  the  first  differ- 
ences), and  so  on. 


216 


EXPLORATORY   SURVEYING. 


Now  n  is  the  ratio  of  the  fractional  period  for  which  we 
wish  to  interpolate,  to  the  interval  between  which  the  values 
are  given  ;  in  this  case  15  minutes  to  1  hour,  therefore  n  =  i  : 
so  that  now  we  have  merely  to  insert  the  upper  values  in  the 
columns  for  di ,  di ,  etc.,  and  the  above  value  of  n,  in  order  to 
find  the  declination  at  2h  15m. 


Dec.  at  2h 

"      4" 

=  18°  17'  4" 

-  7'  59" 
'-8'  05"" 
-  8'  10" 

$2 

=  18°  09'  5" 

-  18°  or  o" 

=  17°  52'  50" 

-  6" 

-5" 

1 


Thus, 


F»  -  18°  17'  4"  -  1'  59".8  -j-  .56"  -  .07"  ; 


therefore, 


Dec.  at  2h  15'"  =  18°  15'  04".75. 


In  such  a  case  as  the  above,  as  it  happens,  the  simple 
method  of  interpolation  would  have  given  Fa  =  18°  15'  04". 2, 
which  of  course  would  have  been  amply  near  enough  for  any- 
thing in  the  way  of  ordinary  work.  But  where  the  explorer  is 
desirous  of  obtaining  a  really  accurate  observation  this  method 
is  often  of  high  value. 

204.  Adjustment  of  Observations.— It  is  a  well-recog- 
nized fact  in  practice,  when  making  a  series  of  measurements 
of  any  quantity,  that  after  every  possible  means  of  eliminating 
and  correcting  for  instrumental  errors  have  been  employed, 
there  still  remain  certain  accidental  errors  which  no  experience 
or  skill  on  the  part  of  the  observer  can  rectify,  since  the  causes 
to  which  they  are  due  are  themselves  unknown.  Thus  it  hap- 
pens that  each  measurement  in  the  set  may  be  different,  al- 
though, judging  from  the  care  taken  in  observing  each  and 
the  apparent  similarity  of  the  conditions  under  which  they 
were  taken,  no  such  differences  should  exist.  The  question 
then  arises  as  to  what  is  to  be  taken  as  the  most  probable 
result. 

Now  according  to  the  Theory  of  Least  Squares,  the  method 
usually  adopted  for  the  solution  of  these  problems,  the  most 
probable  value  of  any  number  of  measurements  of  the  same 
quantity,  each  measurement  being  considered  to  be  equally 
reliable,  is  that  which  makes  the  sum  of  the  squares  of  the 


EXPLORATORY   SURVEYING.  217 

"  errors  "  a  minimum ;  and  the  value  which  does  so  is  the 
arithmetical  mean  of  all  the  measurements.  The  "  error  "  in 
the  case  of  each  measurement  being  its  difference  from  the 
mean. 

But  it  often  happens  that  the  circumstances  under  which  the 
several  measurements  are  made  are  such  as  to  warrant  greater 
"  weight  "  being  given  to  some  of  them  than  to  others.  These 
weights  are  often  deduced  from  the  observations  themselves, 
or  from  them  in  connection  with  a  special  series  of  observa- 
tions ;  but  in  ordinary  field  practice,  weights  assigned  arbitrari- 
ly after  a  thoughtful  perusal  of  all  the  attendant  circumstan- 
ces are  more  likely  to  be  of  value  than  those  found  by  a  strict 
application  of  the  formulas  of  Least  Squares.  Weights  being 
thus  assigned,  the  most  probable  value  of  the  results  will  be 
found  by  multiplying  each  observed  value  by  its  weight,  and 
dividing  the  sum  of  the  products  by  the  sum  of  the  weights, 
the  result  being  that  value  which  renders  the  sum  of  the  prod- 
ucts of  the  squares  of  the  errors  and  the  respective  weights  a 
minimum.  And  this  value  is  termed  the  Weighted  Mean. 
This  may  be  best  illustrated  by  an  example. 

Suppose  that  we  have,  as  several  corrected  measurements  of  a 
base,  the  following  numerators,  and  that,  .considering  all  the 
attendant  circumstances,  we  have  assigned  to  each  the  weight 
shown  as  its  denominator,  assuming,  for  the  sake  of  simplicity, 
that  the  weight  of  the  least  reliable  is  expressed  by  unity: 

2056. 32  feet     ,    2056.20  feet         2056  16  feet 

—  -4-          -3- 

Then  the  most  probable  value  of  the  result  is  given  by 

2056. 32 -+- (2056. 20  X  4)  -f  (2056. 16x3) 
1  +  4  +  3 

A  fair  test  of  precision  in  dealing  with  a  set  of  measure- 
ments is  afforded  by  means  of  the  "  probable  error  "  of  a  sin- 
gle determination,  which  is  found  by  taking  the  difference 
between  each  individual  result  and  the  mean,  squaring  these 
quantities,  and  dividing  their  sum  by  (n  —  1)  where  n  repre- 
sents the  number  of  individual  results  ;  then,  on  extracting 
the  square  root  of  this  quotient  and  multiplying  by  0.674,  we 


218  EXPLORATORY   SURVEYING. 

obtain  the  so-called  Probable  Error.  But  this  terra  does  not 
mean  that  that  error  is  more  probable  than  any  other,  but 
merely  that  in  a  future  observation  the  probability  of  com- 
mitting an  error  greater  than  the  probable  error  is  equal  to 
the  probability  of  committing  an  error  less  than  the  probable 
error. 

The  probable  error  of  the  arithmetical  mean  may  be  simi- 
larly found,  the  value  n(n  —  1)  being  substituted  for  (n  —  1) 
in  the  rule  given  above  for  a  single  determination. 

Errors  in  excess  are  considered  positive  ;  those  in  defect, 
negative. 

205.  Having  now  examined  the  various  methods  of  obtain- 
ing positions  on  exploratory  surveys,  we  next  come  to  the  sub- 
ject of  ascertaining  the  bearings  and  distances  of  these  posi- 
tions relatively  to  each  other  or  to  other  points,  when  taking 
into  consideration  the  curvature  of  the  earth's  surface. 

From  what  has  already  been  said  in  Sec.  58  on  the  subject  of 
the  Convergence  of  the  Meridians,  we  can  see  what  form  the 
corrections  will  have  to  take  in  order  to  allow  for  the  spherical 
— or  more  correctly  spheroidal — form  of  the  earth  ;  and  now, 
by  means  of  3  or  4  simple  problems,  we  can  obtain  all  the 
formulae  necessary  for  the  construction  of  the  groundwork 
of  a  map,  or  the  calculation  of  courses,  which  are  ever  likely 
to  be  needed  in  connection  with  exploratory  surveys. 

In  Engineering  Geodesy  it  is  usually  sufficiently  accurate  to 
assume  the  earth  to  be  a  sphere,  the  radius  of  which  equals 
the  mean  radius  of  curvature  of  the  spheroid  ;  but  it  may  be 
as  well  here  to  examine  the  subject  roughly,  in  order  that  the 
engineer  may  have  an  idea  of  the  extent  of  the  errors  which 
this  assumption  involves. 

206.  THE  FIGURE    OF    THE   EARTH.— According  to 
Col.  Clarke, 

the  mean  Equatorial  semi-axis  =  20926202  feet, 
and  the  Polar  Semi-axis  =  20854895  feet. 

Also  the  radius  of  curvature  in  the  direction  of  the  meridian  in 
any  latitude  L  equals  in  feet 

R  =  20890564  -  106960  cos  2Z+  228  cos  4£; 


EXPLORATORY  SURVEYING. 

and  the  radius  of  curvature  in  a  direction  perpendicular  to  the 
meridian  equals  in  feet 

r  =  20961932  -  35775  cos  2L  +  46  cos  4£. 
Thus  at  the  Equator 

R  =  20783832  feet,    r  =  20926203  feet ; 
and  at  the  poles 

M  =  20890564  feet,     r  =  20961932  feet. 

So  that  for  engineering  purposes  we  may  take  20,890,000 
feet  as  the  mean  radius  of  curvature.  Again,  according 
to  the  same  authority,  the  length  of  a  degree  of  latitude 

equals  in  feet 

D  =  364609.1  -  1866.7  cos  2L  -f  4  cos  4L, 
and  the  length  of  a  degree  of  longitude  equals  in  feet 
d  =  365542.5  cos  L  -  311.8  cos  3£  +  0.4  cos  5£. 

The  value  of  the  foot  taken  above  is  the  English  standard, 
which  is  less  than  the  American  standard  in  the  ratio  of  1  mile 
to  1  mile  and  3.677  inches. 

For  rough  work  we  may  consider 

D  =  364000  feet     and    d  =  D  cos  Lat. 

Table  XVIII  gives  the  true  values  of  1  minute  of  arc,  to  the 
nearest  foot. 

207.  Now  from  the  formula  for  the  length  of  a  circular 
arc  given  in  Sec.  73,  if  we  take  the  above  value  pf  the  mean 
radius  of  curvature,  we  find  the  length  of  an  arc  on  the  earth's 
surface  in  feet  equals 

I  =  6076  n  (nearly), 

where  n  =  the  number  of  minutes  in  the  arc ;  and  the  con- 
verse of  this, 


220  EXPLORATORY   SURVEYING. 

enables  us  to  convert  any  given  distance  into  its  equivalent  in 
augular  measure. 

If  it  is  desirable  to  obtain  the  value  of  I  more  accurately  than 
by  this  means,  we  can  do  so  by  obtaining  first  the  value  of  I 
in  the  direction  of  the  meridian,  either  from  Table  XVIII,  or 
more  correctly  by  dividing  the  value  of  D,  given  in  Sec.  206, 
by  60.  Also  the  length  of  a  1'  arc  perpendicular  to  the  merid- 
ian is  needed,  which  may  be  obtained  by  means  of  the  value  of 
r,  given  in  Sec.  206.  Then  if  we  call  this  latter  value  I',  the 
length  of  an  arc  subtending  1'  at  the  earth's  centre,  which 
makes  an  angle  A  with  the  meridian,  equals 

Zcos2  A  +  l1  sin2  A 

208.  Given  the  latitude  and  longitude  of  two  places 
to  obtain  their  distance  apart,  and  the  bearing  of  the 
course  joining  them. — Suppose  A  and  D  in  Fig.  12  are  the 
two  given  places,  then  the  arc  AF&nd  the  arc  ED  represent 
their  latitudes.  Then  in  the  spherical  triangle  AND,  since  N 
—  difference  of  longitude,  and  AN  and  ND  are  equal  to  the 
co-latitudes  of  A  and  D,  we  can  find  AD  thus: 

cos  AD  =  sin  a  sin  d  -j-  cos  a  cos  d  cos  AND, 

where  a  and  d  are  the  latitudes  of  A  and  D.  And  the  bearing 
of  the  arc  AD,  which  at  A  is  represented  by  the  angle  NAD, 
is  then  given  by  the  equation 

sin  A  =  cos  d  cosec  AD  sin  AND. 
Or,  if  A  and  D  are  in  the  same  latitude,  we  have 
tan  A  =  cot  %AND  cosec  lat. 

The  arc  so  obtained  can  be  converted  into  feet  as  shown  in 
Sec.  207;  and  this  is  the  distance  along  the  arc  of  the  great 
circle  passing  through  A  and  D,  i.e.,  the  shortest  distance  be- 
tween them  on  the  earth's  surface. 

Conversely,  given  the  latitude  and  longitude  of  A, 
and  the  bearing  and  distance  of  another  place  J),  to  find 
the  latitude  and  longitude  of  J).— First  convert  AD  into 
angular  measure  according  to  Sec.  207;  then  we  have  the  sides 


EXPLORATORY    SURVEYING.  221 

AD,  AN,  and  the  included  angle  A.    Then  to  find  d  we  have 

sin  d  =  cos  AD  sin  a  -j-  sin  AD  cos  a  cos  A 
Then  AND,  the  difference  of  longitude,  is  given  by 

sin  AND  =  sin  A  sin  AD  sec  d. 

The  bearing  of  AD  at  D  may  be  obtained  from  the  equation 
sin  D  =  sin  J..ZV.D  cos  a  cosec  AZ). 

The  formulae  given  in  this  section  are  simply  those  ordinarily 
used  for  the  solution  of  spherical  triangles.     (See  Sec.  233.) 

209.  To  find  the  radius  of  a  Circle  of  Latitude.— In 
Fig.  93  let  C  be  the  centre  of  the  earth,  N  the  P 
pole,  and  L  any  given  latitude;  then,  consider- 
ing the  earth  to  be  a  sphere,  the  angle  LPG  = 

the  latitude  of  L,  so  that 

PL  =  LC  cot  latitude, 

where  PL  =  radius  of  the  circle  of  latitude. 
LC  may  be  taken  as  equal  to  20,890,000  feet. 

210.  To    calculate  the    offset  at  aiiy 
point  C  to  a  parallel  of  latitude  AC 
from    a    straight  Hue  AB,  tangent    to 
AC  at  A. — We  can  do  this  by  treating  the 

parallel  of  latitude  A  C  in  Fig.  94  as  a  curve          FIG.  93. 
to  which  the  arc  of  a  great  circle  AB  is  tangent  at  A,  and  thus 
obtain  the  offset  CB  according  to  Sec.  78;  or,  we  can  solve  the 
N  right-angled  spherical   triangle  ANB,  and   so 

find  the  latitude  of  B,  if  we  know  the  differ- 
ence of  longitude  N,  thus: 

tan  (lat.  B)  —  tan  (lat.  A)  cos  N. 
CB  then  equals  the  difference  of  latitude  of  A 
and  B. 

211.  We  are  now  in  a  position  to  consider 
the  influence  of  the  spherical  form  of  the  earth, 
assuming   for  the   moment  the  earth  to  be  a 
sphere,  on  a  map  the  linear  measurements  of 
which  have  been  computed  on  the  supposition  that  the  sur- 
face of  the  earth  is  a  plane, 


222          EXPLORATORY  SURVEYING. 

Now  a  spherical  surface  cannot  be  developed  on  a  plane 
surface,  but  can  only  be  developed  on  a  sphere  of  equal  radius. 
Thus  no  map  can,  theoretically  even,  be  correct  to  the  same 
scale  in  all  its  parts.  In  nautical  charts,  which  are  gen- 
erally made  on  Mercator's  Projection,  this  difficulty  is  over- 
come by  the  use  of  a  scale  of  meridional  parts,  the  scale  at 
all  points  being  proportional  to  the  secant  of  the  latitude. 
And  this  is  a  very  convenient  method,  where  all  positions  are 
obtained  astronomically  and  where  the  error  involved  by 
calculating  the  courses  according  to  "  Middle  Latitude  Sail- 
ing" is  of  no  importance.  But  in  constructing  a  map  this 
method  is  inconvenient;  for  if  the  same  scale  is  used  through- 
out, it  assumes  that  parallels  of  latitude  are  right  lines,  and 
that  there  is  no  convergence  of  the  meridians.  In  plotting 
exploratory  surveys,  simplicity  is  an  important  factor;  also, 
the  map  must  be  adapted  to  the  same  scale  throughout,  and 
be  so  arranged  as  to  be  suitable  to  the  plotting  of  topography 
as  on  a  plane  surface.  To  approximate  as  near  as  possible  to 
correctness  in  the  more  important  portions,  and  to  throw  the 
excess  of  error  into  the  less  important  parts,  is  the  best  that 
can  be  done  under  any  circumstances. 

212.  In  Sec.  58  we  referred  to  the  corrections  which  it  was 
necessary  to  make  on  account  of  the  convergence  of  the  merid- 
ians. By  extending  this  method  we  are  able,  with  the  aid  of 
the  preceding  problems,  to  construct  the  groundwork  of  our 
map  without  any  other  principles  than  those  already  explained. 
The  best  way  is  to  take  an  example  and  work  it  out  as  if  in 
actual  practice. 

Suppose  from  A  in  Latitude  60°  N.  and  Longitude  120°  W. 
we  intend  starting  off  straight  across  country  for  B,  a  place 
which,  from  the  maps,  we  find  to  be  situated  in  about  Lat.  59° 
N.  and  Long.  110°  W.,  and  wish  before  starting  to  lay  out  the 
groundwork  of  a  map  to  be  constructed  from  the  knowledge 
of  the  topography  which  we  intend  to  obtain  on  the  way— that 
we  may  have  some  reliable  means  of  plotting  our  results  as 
soon  as  obtained,  and  also  of  determining  positions  relatively 
to  each  other  by  means  of  bearings  and  distances. 

At  A  we  draw,  as  in  Fig.  95,  the  base-lines  AS  and  AD. 
Then  find  the  length  of  AC  from  Table  XVIII,  calculating 
as  if  it  were  in  the  mean  latitude  of  A  and  B,  i.e.,  59°  30' 
N. ;  thus  AC  =  about  10  X  60  X  3095  =  say  1,857,000  feet.  If 


EXPLORATORY    SURVEYING. 


223 


great  accuracy  were  required,  we  could  find  the  value  of  d  in 
latitude  59°  30'  according  to  Sec.  206,  then  AC  =  IQd. 

-t    N-'^-^"^ 
g 


FIG.  95. 

Next  we  make  AD  =  AC,  and  through!)  draw  the  meridian 
CB,  the  bearing  of  which  on  the  map,  relatively  to  A,  =  the 
convergence  between  A  and  B  =  8"  36'.  Therefore  the  angle 
CD  A  =  81°  24'. 

The  length  of  the  offset  CD  may  be  found  according  to  Sec. 
78,  and  is  equal  to  about  140,000  feet;  and  since  B  lies  1°  to  the 
south  of  C,  and  on  the  meridian  passing  through  D,  we  have 
DB  =  about  225,400  feet.  Then  by  solving  the  plane  triangle 
ADB,  we  obtain  AB  =  1,903,800  feet,  and  -the  angle  BAD  = 
6°  44'.  Thus  the  direct  course  from  A  to  B  is  S.  83°  16'  E., 
and  Ad  =  "  Total  departure"  =  AB  cos  6°  44'  —  1,890,700  feet, 
and  Bd  —  "  Total  latitude"  =  DB  cos  8°  36'  =  222,800  feet. 

We  have  thus  the  groundwork  of  our  map  ready  for  the 
plotting  of  the  courses,  and  if  we  use  sheets  of  cross-section 
paper,  with  10  divisions  to  the  inch,  and  plot  to  a  scale  of 
10,000  feet  to  an  inch,  we  then  have  a  map  of  tolerably 
convenient  size,  plotted  to  a  scale  sufficiently  large  to  show 
the  main  features  of  the  country,  since  any  important  parts 
which  may  have  been  made  the  subjects  of  special  survey  can 
be  best  shown  separately. 

In  order  to  connect  the  Astronomical  work  with  that  which 
is  plotted  by  Latitudes  and  Departures,  or  by  protractor, 
and  which  we  may  call  our  "  dead-reckoning,"  we  must  draw 
meridians  and  curves  of  latitude  at  about  every  30'.  To  fill 
in  these  meridians,  divide  A  C  equally  into  20  parts,  and  draw 
the  meridians  perpendicular  to  the  curve  at  each  of  these 
points,  i.e.,  dividing  up  the  convergence  equally  among 
them.  The  curve  of  latitude  AC,  since  we  know  the  dis- 
tance CD,  can  be  drawn  by  assuming  that  the  offset  half-way 
between  A  and  D  =  %CD,  and  so  on,  according  to  Sec.  78. 


224  EXPLORATORY   SURVEYING. 

The  advantages  of  this  method  of  plotting  are,  that  we  can 
readily  connect  positions  taken  by  astronomical  observations 
with  those  calculated  from  dead-reckoning,  the  former  being 
plotted  by  the  guidance  of  the  parallels  of  latitude  and  the 
meridians,  and  the  latter  by  means  of  the  base  Ad.  Also,  that 
the  same  scale  is  used  throughout,  and  the  bearings  of  all 
points  may  be  taken  off  with  a  protractor. 

If  the  topographical  positions  are  obtained  solely  by  direct 
astronomical  observations,  tJien  the  method  of  Mercator's  Pro- 
jection is  more  convenient  than  that  given  above. 

To  plot  our  route  we  proceed  as  follows:  Suppose  we  take 
rough  compass  courses;  these  we  plot  lightly  on  the  map, 
having  worked  them  out,  say,  by  Latitudes  and  Departures, 
correcting  the  "  latitudes"  absolutely  according  to  any  latitude 
observations  we  may  take,  the  "  departures"  being  guided  to  a 
reasonable  extent  by  the  observations  for  longitude.  Thus  our 
course  is  constantly  being  broken,  involving  a  new  "total 
latitude"  for  each  fresh  start.  This  we  can  best  find  by  scal- 
ing from  Ad,  after  having  plotted  the  position  astronomically. 
At  the  end  of  our.  journey,  whatever  error  in  longitude  we 
may  have,  may  usually  be  divided  up  proportionally  along  the 
whole  route,  if  the  trip  has  been  made  at  a  tolerably  uniform 
pace.  The  error  in  latitude  should  be  inappreciable. 

The  above  example  shows  what  must  be  considered  in  plot- 
ting an  extensive  survey;  and  though  a  more  rough  and  ready 
method  is  usually  correct  enough,  yet  where  the  field-work  is 
run  in  such  a  way  as  to  warrant  a  tolerably  accurate  plot  of  it 
being  made,  the  little  extra  time  involved  in  making  a  good 
map  is  time  well  spent. 

As  regards  the  mode  of  procedure  in  keeping  a  course  astro- 
nomically, Col.  Frome  says:  "It  is  probably  inconvenient 
always  to  obtain  latitude  at  noon,  but  we  can  generally  do  so, 
and  more  correctly,  at  night  by  the  meridian  altitude  of  one 
or  more  of  the  stars.  The  local  time  can  immediately  before 
or  after  be  ascertained  by  a  single  altitude  of  any  other  star 
out  of  the  meridian — the  nearer  the  prime  vertical  the  better; 
and  if  a  pocket-chronometer  is  carried,  upon  which  any  de- 
pendence can  be  placed,  the  explorer  has  thus  the  means,  by 
comparison  with  his  local  time,  of  obtaining  his  approximate 
longitude,  and  laying  down  his  position  on  paper.  The  lon- 
gitude should  also  be  obtained  occasionally  by  Lunar  Dis- 


EXPLORATORY   SURVEYING. 


225 


tances,  or  some  other  method.  The  latitude  he  should  always 
get  correct  to  half  a  mile,  and  the  longitude  to  8  or  10  miles." 
213.  The  Star  Map  given  below  will  be  found  convenient  in 
selecting  suitable  stars  for  observations.  The  stars  are  plotted 
from  their  R.A.'s  and  Decs,  in  the  same  way  that  a  map  of  the 
earth  is  plotted  by  longitudes  and  latitudes,  i.e.,  looking  down 
on  it. 


STAR    MAP 

FOR *" 

NORTHERN  HEMISPHERE. 

The  centre  is  the  celestial  pole,  and  the  24  radiating  lines 
divide  the  24  hours  of  R.A.  Now  the  initial  point  for  R.A. 
being  on  the  meridian  at  10  P.M.  about  Oct.  21,  we  can  divide 
the  circle  into  12  divisions,  and  arrange  them  so  that  the  radi- 
ating line  marked  0  Hours  will  cut  the  10  o'clock  division 
about  two  thirds  along  it.  Thus  we  read  off  that  about  Oct.  21 
the  star  marked  1  will  be  on  the  meridian,  i.e.,  due  south,  at 


226 


EXPLORATORY   SURVEYING. 


10  P.M.  Similarly  the  star  marked  23  will  be  on  the  meridian 
at  10  P.M.  about  Aug.  17. 

But  suppose  we  want  to  know  what  star  will  be  near  the 
meridian  about  8  P.M.  on  Jan.  10.  Imagine  the  margin  of  the 
map,  with  the  months  marked  on  it,  to  be  stationary,  and  the 
interior  portion  to  rotate  in  the  same  direction  as  the  hands  of 
a  watch,  once  in  23h  56m  ;  then,  since  the  map  shows  the  posi- 
tion at  10  p.m.,  at  8  P.M.  (two  hours  earlier)  the  star  marked 
5  will  have  been  near  the  meridian  on  Jan.  10, 

In  this  way  we  can  tell  at  about  what  time  any  meridian  ob- 
servation will  occur  without  referring  to  the  Nautical  Almanac, 
Thus  with  this  map  and  the  following  key  and  table  no  Nauti- 
cal Almanac  is  needed  for  latitude  observations,  by  the  merid- 
ian altitudes  of  stars.  The  Decs,  and  R.A.'s  given  are  for 
Jan.  1,  1889. 


TABLE  OF  MAGNITUDE,   DEC.,  AND  R.A.   OF  THE 
PRINCIPAL  STARS. 


No. 
in 
Map. 

NAME. 

i 

Dec. 

An. 
Var. 

R.A. 

An. 
Var. 

0        1        II 

„ 

h.  m.  s. 

s. 

1 

Alpherat,  a  Andromedse 

2.0 

-  -  28  28  39 

+  19.88 

0    2  39 

+  3.09 

2 

Polaris,  a  Ursae  Minoris.. 

2.0; 

-  -  88  42  59 

+  18.90 

1  18  08 

+  23.15 

3 

•y  Cassiopeia?  

2.0 

-  -  60  06  55 

4-  19.56 

0  50  01 

r3.58 

4 

Algol   /3  Persei  

2.7 

-  -4031  38  +  14.1'2 

3    0  57 

-3.88 

5 

a  Persei 

2.0 

-  -49  27  55;  4-  13.10 

3  16  24 

-4  26 

6 

Aldebaran,  a  Tauri  

1.0 

-  -  16  17  07 

+  7.52 

4  29  33 

-3.44 

7 

Capella,  a  Aurigae  

l.O1 

-  -  45  53  03 

4-  4.03 

5  08  29 

-442 

8 

a  Arietis  

2.0: 

--2256  14 

+  17.7 

2    0  55 

-  3.37 

9 

Rigel,  £  Orionis  

1.0 

-81950 

+  4.40 

5  09  12 

-  2.88 

10 

Betelgeuze,  a  Orionis... 

1.2 

4  7  23    8 

+  0.95 

5  49  10 

-  3.25 

11 

Sirius,  a  Canis  Majoris.. 

1.0 

-  16  33  5-2 

-  4.71 

6  40  15 

-2.64 

12 

Castor,  a  Geminorum.  .  . 

1.7 

+  32  07  53 

-  7.55 

7  27  31 

-3.84 

13 

Pollux,  /3  Geminorum... 

1.3 

4-281737 

-8.41 

7  38  31 

-368 

14 

Procyon,  a  Canis  Minoris 

1.0 

4-  5  30  3-2 

-  8.99 

7  33  29 

-3.14 

15 

Regulus,  a  Leonis.  

1.3  4-'  12  30  34 

-  17.47110  02  28 

-320 

16 

a  Ursae  Majoris  

2.04-6221    0 

-  19.36  10  56  52 

-  3.75 

17 

y  Ursae  Majoris  

2.34-54  1842 

-  20.03  11  47  59 

-  3.18 

18 

TJ  Ursae  Majoris  

2.0,4-495203 

-  18.08'  13  43  10 

-  2.37 

19 

Arcturus,  a  Bootis  

1.01+  19  45  38 

-  18.88J14  10  36 

-  2.73 

20 

Spica,  a  Virginis  

1.0  -  103454 

—  18.90  13  19  21 

-3.15 

21 

AntT/res    a  Scorpii 

1.3 

-  26  11  06 

—  8.30  16  22  36 

-  3.  (57 

22 
23 
24 
25 

Vega,  a  Lyrae  

1.0 
1.3 

1  7 

4-384050     4-3.1? 
4-  8  34  32l    +9.27 
4  44  53  02  j  +  12.72 
-  301237  4-18.99 

18  33  11 
19  45  22 
20  37  39 
22  51  31 

-  2.03 
-  2.93 
-204 
-  3.8-2 

Altair.  a  Aquilee  
a  Cvgni 

Fomalhaut,  a  P.  Aust.  .  . 

1.3 

26 

Mar  kiib    a  Pegasi. 

2.0 

+  14  36  29 

+  19.30  22  59  14 

-2.98 

EXPLORATORY   SURVEYING.  227 

IK  THE  SOUTHERN  HEMISPHERE  WE  ALSO  HAVE- 


NAME. 

Mag. 

Dec. 

An.  Var. 

R.A. 

An. 
Var. 

/3  Hydri 

3  0 

77  52  46 

4-2028 

h.  m.  s. 
0  19  54 

s. 
4-323 

Achernar,  a  Eridani  .. 
Canopus,  a  Argus  
ft  Argus  
a  Crueis 

1.0 
1.0 
1.5 
1  0 

-  57  48  03 
-  52  38  07 
-  69  15  36 
-  62  29  02 

4-  18.36 
-  1.87 
-  14.80 
-  2001 

1  33  M 
6  21  29 
9  11  59 
12  20  26 

-1-2.23 
4-  1.83 
+  0.68 
-  3  29 

/3  Centauri  

1.0 

-  59  50  14 

—  17  59 

13  55  59 

--4-18 

a  Centauri  
a  Trianguli  Aust  

1.0 
2.0 
2  0 

-  60  22  47 
-  68  49  21 
_j_  12  33  29 

-  15.38 
-  7.16 
2  87 

14  32  05 
16  36  55 
17  29  47 

-  -  4.05 
-  -  6.30 

-  -  2  78 

a  Gruis  

2.0 

—  47  29  53 

4-  17.25 

22  01  14 

--3  81 

In  order  better  to  recognize  the  positions  of  the  stars  at  night, 
they  may  be  pricked  through  on  a  sheet  of  paper,  which, 
when  turned  backwards  and  held  up  towards  the  south,  with  the 
month  at  the  lowest  part,  will  correspond  with  the  face  of  the 
sky  at  10  P.M. 


PART   IV. 
MISCELLANEOUS. 


THE  following  miscellaneous  information  may  at  times  be 
found  of  service  in  the  field  to  both  the  engineer  and  the  ex- 
plorer: 

2U.  To  find  the  Horse-power  of  Falling  Water. 

H.P.  =  0.00189  QH, 

where  Q  =  the  number  of  cubic  feet  of  water  passing  over  the 
fall  per  minute,  and  H=  height  of  fall  in  feet. 

Turbines  can  utilize  about  75  p.  c.  of  this  H.P.  Thus  the 
Effective  horse-power,  i.e.,  available  for  useful  work,  =  about 
.0014  QH. 

215.  To  gauge  a  stream,  roughly.  Take  some  body, 
which,  when  floating,  will  be  almost  entirely  immersed,  and 
throw  it  into  the  middle  of  the  stream,  in  a  part,  if  possible, 
unobstructed  by  reeds,  etc.,  and  free  from  slack-  water,  eddies, 
or  counter-currents;  and  where  the  cross-section  of  the  stream 
is  fairly  uniform.  Observe  the  time  T  in  seconds  which  the 
body  takes  to  float  a  distance  of  100  feet.  Then  if  A  =  the 
cross-section  of  the  stream  in  square  feet,  and  Q  =  cubic  feet  of 
water  that  pass  per  minute, 

5000J. 


This  assumes  that  the  middle  surface  velocity  is  to  the  mean 
velocity  as  6  to  5,  which  is  a  fairly  average  ratio. 

228 


MISCELLANEOUS.  229 

216.  The  Sustaining  power  of  ordinary  wooden  piles 

in  Ibs.  equals 

FW 

-IT 

where 

F  —  fall  of  hammer  in  inches, 
W=  weight  of  hammer  in  Ibs., 
S  =  space  driven  by  last  blow  in  inches. 

This  formula  is  generally  found  to  give  results  about  as  re- 
liable as  any  general  formula  can  give. 

217.  Supporting  power  of  various  materials. 

Clay 1.0  to  2.0  tons  per  sq.  foot. 

Sandy  clay 2.0to4.0    " 

Sand 3.0  to  5.0    "  " 

Gravel 4.0  to  5.0    " 

Sandstone 2.0  to  4,0    " 

Firm  Rock 10.0    ' ' 

These  are  the  pressures  to  which  the  above  may  usually  be 
safely  loaded. 

218.  Transverse  strength  of  rectangular  beams. 
Let  L  =  length  of  beam  in  feet  between  points  of  support, 

b  =  breadth  of  beam  in  inches, 
d  =  depth  of  beam  in  inches, 
W=  Load  at  centre  of  beam  in  Ibs., 
/=  coefficient  of  modulus  of  rupture. 
Then 


1SWL  1       A 

''  and  6= 


For  the  values  of  /see-following  table. 

For  example,  if  b  =  6",  d  =  W",  and  L  —  20  feet,  if  we 
take/=  10,000  Ibs.,  by  the  above  formula  W=  16,666  Ibs.  ;  so 
that  with  a  Factor  of  Safety  of  6  we  may  safely  load  it  at  its 
centre,  and  consequently  at  any  part  of  it,  with  a  weight  of 
2778  Ibs. 

A  beam  will  carry  as  a  centre  load  only  half  the  weight  that 
it  will  bear  distributed  uniformly  over  it.  So  that,  for  instance, 
if  we  wish  to  know  what  total  breadth  we  must  give  to  a  set 
of  stringers,  where  d  =  16"  ,  in  order  safely  to  carry  an  ordinary 
train  over  a  span  of  15  feet,  if  we  take/=  10,000  Ibs.  and  the 


230 


MISCELLANEOUS. 


load  per  foot  run  as  equivalent  to  4000  Ibs.,  we  have  as  the 
equivalent  value  of  W,  30,000  Ibs.  So  that  by  the  above 
formula  b  =  about  3  inches.  Therefore,  taking  a  factor  of 
safety  of  8,  b  =  about  24  inches;  so  that  four  6"  X  16"  stringers 
may  safely  be  used.  The  factor  of  safety  usually  adopted  for 
wood  varies  from  5  to  10,  according  to  the  condition  of  the 
timber,  the  amount  of  impact  caused  by  the  load,  and  the 
possible  amount  of  decay  to  which  it  will  be  subjected. 

For  spans,  in  railroad  bridges,  less  than  10  feet,  5000  Ibs.  per 
foot  run  should  usually  be  taken  as  the  uniformly  distributed 
load.  In  spans  exceeding  15  feet  3500  Ibs.  is  usually  sufficient. 
These  values  take  no  account  of  the  weight  of  the  beams  them- 
selves. 

VALUES  OF  /. 


Material. 

Lbs.  per  sq.  in. 

Material. 

Lbs.  per  sq.  in. 

Ash 

12,000  to  14,000 

Red  Pine  .  .  . 

7100  to  9500 

Birch  

11,700 

Spruce  .  .  . 

9900  to  12,300 

Blue  Gum  
Elm 

18,000 
6000  to  9700 

Brit.  Oak  
Am.  Red  Oak 

12,000 
10600 

219.  Natural  Slopes  of  Earths. 


Material. 

1 

53 

Material. 

! 

53 

Material. 

i 

o 

02 

Gravel 

40° 

Vegetable  Earth 

28° 

Ruble 

45° 

Dry  Sand  
Sand 

38° 

goo 

Compact  Earth  .  .  . 
Shingle 

50° 
39° 

Clay  (drained)  
Clay  (wet) 

45° 
16° 

220.  Weight  of  Earths,  Kocks,  etc.,  per  cubic  yard, 


Material. 

Weight 
in  Ibs. 
per 
cu.  yd. 

Material. 

Weight 
in  Ibs. 
per 
cu.  yd. 

Material. 

Weight 
in  Ibs. 
per 
cu.  yd. 

Sand  
Gravel  
Mud  
Marl  

3360 
3360 
2800 
2900 

Clay  
Chalk  
Sandstone.  . 
Shale  

3470 
4030 
4370 
4480 

1 
;  Quarts  
Granite  .  .  . 
Trap  
;  Slate.  

4590 
4700 
4700 
4810 

A  cubic  yard  of  water  weighs  about  1680  Ibs. 


MISCELLANEOUS. 


231 


221.  Weight  of  Timber  and  Metals  per  cubic  foot. 


Material. 

Weight 
in  Ibs. 
per 
cu.  ft. 

Material. 

Weight 
in  Ibs. 
per 
cu.  ft. 

Material. 

Weight 
in  Ibs. 
per 
cu.  ft. 

Elni,  English  . 
Canadian  Elm 
Maple 

35 
45 
42 

Pine,  red.  ... 
"     white  .  . 
Teak  

36 
30 
50 

Iron,  cast  — 
"    wrought 
Steel  

450 

482 
490 

English  Oak  .  . 
American  Oak 

48 
50 

Spruce  
Larch  

30 
34 

Copper  
Lead  

550 
710 

222.  Mortar,  Cement,  etc.  (common  mixtures). 

Mortar.— \  of  lime  to  2  or  3  of  sharp  river  sand. 

Coarse  Mortar. — 1  of  lime  to  4  of  coarse  gravelly  sand. 

Concrete. — 1  of  lime  to  4  of  gravel  and  2  of  sand. 

Hydraulic  Mortar. — 1  of  blue  lias  lime  to  2£  of  burnt  clay, 
ground  together. 

Beton. — 1  of  hydraulic  mortar  to  H  of  angular  stones. 

Cement. — 1  of  sand  to  1  of  cement;  or  if  great  tenacity  is 
required  the  sand  may  be  omitted. 

Portland  Cement  is  composed  of  clayey  mud  and  chalk 
ground  together  and  afterwards  calcined  at  a  high  temperature, 
and  then  ground  to  a  fine  powder. 

NOTES. — For  ordinary  engineering  work  the  following 
proportions  make  a  good  mortar  : 
1  measure  of  Lime; 

3  to  5  measures  of  sand,  according  to  the  "  hunger  "  of  the  sand, 
1  measure  of  ashes,  brick  dust,  or  burnt  clay. 

For  engineering  work,  if  exposed  to  dampness,  £of  the  lime 
in  the  above  should  be  replaced  by  hydraulic  cement ;  whilst 
for  work  under  water,  1  measure  hydraulic  cement  to  2 
measures  of  sand  make  a  good  mixture. 


NOTES  ON  TIMBER. 

223.  Selection  of  standing  trees.  —  "  Scribner's  Log 
Book." — "The  principal  circumstances  which  affect  the 
quality  of  growing  trees  are  soil,  climate,  and  aspect. 

"In  a  moist  soil  the  wood  is  less  firm,  and  decays  sooner  than 
in  a  dry,  sandy  soil ;  but  in  the  latter  the  timber  is  seldom 
fine  :  the  best  is  that  which  grows  in  a  dark  soil,  mixed  with 


232  MISCELLANEOUS. 

stones  and  gravel.  This  remark  does  not  apply  to  the  poplar, 
willow,  cypress,  and  other  light  woods  which  grow  best  in 
wet  situations. 

"Trees  growing  in  the  centre  of  a  forest  or  on  a  plain  are 
generally  straighter  and  more  free  from  limbs  than  those 
growing  on  the  edge  of  the  forest,  in  open  ground,  or  on  the 
sides  of  hills ;  but  the  former  are  at  the  same  time  less  hard. 
The  toughest  part  of  a  tree  will  always  be  found  on  the  side 
next  the  north.  The  aspect  most  sheltered  from  prevalent 
winds  is  generally  most  favorable  to  the  growth  of  timber. 
The  vicinity  of  salt  water  is  favorable  to  the  strength  and 
hardness  of  white  oak. 

"The  selection  of  timber  trees  should  be  made  before  the  fall 
of  the  leaf.  A  healthy  tree  is  indicated  by  the  top  branches 
being  vigorous,  and  well  covered  with  leaves**,  the  bark  is 
clear,  smooth,  and  of  a  uniform  color.  If  the  top  has  a  reg- 
ular, rounded  form  ;  if  the  bark  is  dull,  scabby,  and  covered 
with  white  and  red  spots,  caused  by  running  water  or  sap, — the 
tree  is  unsound.  The  decay  of  the  uppermost  branches  and 
the  separation  of  the  bark  from  the  wood  are  infallible  signs 
of  the  decline  of  the  tree." 

224.  Defects  of  Timber  Trees  (especially  of  oak).— "Sap, 
the  white  wood  next  to  the  bark,  which  very  soon  rots, 
should  never  be  used,  except  that  of  hickory.  There  are 
sometimes  found  rings  of  light-colored  wood  surrounded  by 
good  hard  wood;  this  may  be  called  the  second  sap  :  it  should 
cause  the  rejection  of  the  tree. 

"Brash-wood  is  a  defect  generally  consequent  on  the  decline 
of  the  tree  from  age  ;  the  pores  of  the  wood  are  open,  the 
wood  is  reddish-colored,  it  breaks  short  without  splinters, 
and  the  chips  crumble  to  pieces. 

"  Wood  which  has  died  before  being  felled  should  in  general  be 
rejected  ;  so  should  knotty  trees,  and  those  which  are  covered 
with  tubercles,  etc. 

"Twisted  wood,  the  grain  of  which  ascends  in  a  spiral  form,  is 
unfit  for  use  in  large  scantling  ;  but  if  the  defect  is  not  very 
decided,  the  wood  may  be  used  for  naves,  and  for  some  light 
pieces 

"  Splits,  checks,  and  cracks,  extending  towards  the  centre,  if 
deep  and  strongly  marked,  make  the  wood  unfit  for  use,  un- 
less it  is  intended  to  be  split. 


MISCELLANEOUS.  233 

"  Wind-shakes  are  cracks  separating  the  concentric  layers 
of  wood  from  each  other;  if  the  shake  extends  through  the 
entire  circle,  it  is  a  ruinous  defect." 

225.  Felling   Timber.— "  The  most  suitable  season  for 
felling  timber  is  that  in  which  vegetation  is  at  rest,  which  is 
the  case  in  midwinter  and  in  midsummer;  recent  opinions 
derived  from  facts  incline  to  give  preference  to  the  latter  sea- 
son.    The  tree  should  be  allowed  to  attain  its  full  maturity 
before  being  felled;  this  period  in  oak  timber  is  generally  at 
the  age  of  from  75  to  100  years,  or  upwards,  according  to  cir- 
cumstances.   The  age  of  hardwood  is  determined  by  the  num- 
ber of  rings  which  may  be  counted  in  a  section  of  the  tree. 

"The  tree  should  be  cut  as  near  the  ground  as  possible,  the 
lower  part  being  the  best  timber.  The  quality  of  the  wood  is 
in  some  degree  indicated  by  the  color,  which  should  be 
nearly  uniform  in  the  heart  wood,  a  little  deeper  toward  the 
centre,  and  without  transitions. 

"  Felled  timber  should  be  immediately  stripped  of  its  bark, 
and  raised  from  the  ground. 

"  As  soon  as  practicable  after  the  tree  is  felled  the  sap-wood 
should  be  taken  off  and  the  timber  reduced,  either  by  sawing 
or  splitting,  nearly  to  the  dimensions  required  for  use. 

"  The  best  method  of  preventing  decay  is  the  immediate  re- 
moval of  it  to  a  dry  situation,  where  it  should  be  piled  in  such 
a  manner  as  to  secure  a  free  circulation  of  air  around  it,  but 
without  exposure  to  the  sun  and  wind.  .When  thoroughly 
seasoned  before  cutting  it  up  into  small  pieces,  it  is  less  liable 
to  warp  and  twist  in  drying.  When  green,  timber  is  not  so 
strong  as  when  thoroughly  dry. 

"  Lumber  containing  much  sap  is  not  only  weaker,  but  de- 
cays much  sooner  than  that  free  from  sap." 

226.  Seasoning  and  Preserving  Timber.— "For  the  pur- 
pose of    seasoning,  timber  should  be  piled  under  shelter, 
where  it  maybe  kept  dry,  but  not  exposed  to  a  strong  current 
of  air;  at  the  same  time  there  should  be  a  free  circulation  of 
air  about  the  timber,   with  which  view  slats  or  blocks  of 
wood  should  be  placed  between  the  pieces  that  lie  over  each 
other,  near  enough  to  prevent  the  timber  from  bending.    The 
seasoning  of  timber  requires  from  two  to  four  years,  accord- 
ing to  its  size. 


234  MISCELLANEOUS. 

"  Gradual  drying  and  seasoning  in  this  manner  is  considered 
the  most  favorable  to  the  durability  and  strength  of  timber. 

"  Timber  of  large  dimensions  is  improved  by  immersion  in 
water  for  some  weeks.  Oak  timber  loses  about  one  fifth  of 
its  weight  in  seasoning,  and  about  one  third  of  its  weight  in 
becoming  dry." 

227.  Decay  of  Timber.— There  are  three  principal  causes 
of  decay  of  timber — dry-rot,  wet-rot,  and  the  "  teredo  navalis" 
and  other  worms. 

Dry-rot  does  not  usually  occur  where  there  is  a  free  circu- 
lation of  air,  and  if  the  timber  is  properly  dried  an  occasional 
immersion  in  water  should  do  no  harm.  Timber  kept  dry 
and  well  ventilated  has  been  known  to  last  for  several  hun- 
dred years  without  apparent  deterioration.  Dry-rot  is  caused 
by  a  species  of  wood  fungus — Merulius  lacJirymans — which 
destroys  the  tensile  and  cohesive  strength,  gradually  convert- 
ing the  timber  into  a  fine  powder. 

Wet-rot.—  This  is  the  destructive  agent  at  work  more  or  less 
on  all  timber  freely  exposed  to  air  and  moisture.  It  is  of  two 
kinds  : 

A.  CJiemical. — In  this  case  a  slow  combustion  takes  place, 
and  by  a  gradual  process  of  oxidation  the  wood  slowly  rots 
away. 

B.  Mechanical. — This  is  the  more  common  form,  and  gener- 
ally occurs  near  the  water-line  in  timber  subject  to  frequent 
immersion.    It  is  the  frequent  alternate  conditions  of  moisture 
and  dryness  that  are  most  trying  to  timber,  as  is  the  case  with 
metals.    AVhen  timber  is  constantly  under  water,  the  action  of 
the  water  dissolves  a  portion  of  its  substance,  which  is  made 
apparent  by  its  becoming  covered  with  a  coating  of  slime,  and 
this  protects  the  interior.     If,  however,  it  is   exposed  to  al- 
ternations of  moisture  and  dryness,  as  is  the  case  with  piles  in 
tidal  waters,  the  dissolved  parts  being  continually  removed  by 
evaporation  and  the  action  of  the  water,  new  surfaces  are  be- 
ing frequently  exposed  for  decomposition. 

Piles  driven  in  sea-water  are  frequently  destroyed  by  the 
"  teredo  navalis,"  and  also  by  another  species  of  worm  called 
the  "  limnoria."  They  both  work  from  about  the  high-water 
mark  to  the  surface  of  the  mud. 

228.  To    test  Steel  and  Iron.  —  Scientific   American. — 
Nitric  acid  will  produce  a  black  spot  on  steel;  the  darker  the 


MISCELLANEOUS.  235 

spot  the  harder  the  steel.     Iron,  on   the  contrary,  remains 
bright  if  touched  with  nitric  acid. 

Good  steel  in  its  soft  state  has  a  curved  fracture  and  a  uni- 
form gray  lustre;  in  its  hard  state,  a  dull,  silvery,  uniform 
white.  Cracks,  threads,  or  sparkling  particles  denote  bad 
quality. 

Good  steel  will  not  bear  a  white  heat  without  falling  to 
pieces,  and  will  crumble  under  the  hammer  at  a  brig7it-rcd 
heat,  while  at  a  middling  heat  it  may  be  drawn  out  under  the 
hammer  to  a  fine  point.  Care  should  be  taken  that  before  at- 
tempting to  draw  it  out  to  a  point  the  fracture  is  not  concave; 
and  should  it  be  so,  the  end  should  be  filed  to  an  obtuse 
point  before  operating.  Steel  should  be  drawn  out  to  a  fine 
point  and  plunged  into  cold  water;  the  fractured  point  should 
scratch  glass.  To  test  its  toughness,  place  a  fragment  on  a 
block  of  cast-iron:  if  good,  it  may  be  driven  by  a  blow  of  a 
hammer  into  the  cast-iron;  if  poor,  it  will  crush  under  the 
blow. 

Tests  of  Iron. — A  soft  tough  iron,  if  broken  gradually, 
gives  long  silky  fibres  of  leaden-gray  hue,  which  twist  to- 
gether and  cohere  before  breaking. 

A  medium  even  grain  with  fibres  denotes  good  iron. 
Badly  refined  iron  gives  a  short  blackish  fibre  on  fracture.  A 
very  fine  grain  denotes  hard  steely  iron,  likely  to  be  cold- 
short and  hard. 

Coarse  grain  with  bright  crystallized  fracture  or  discolored 
spots  denotes  cold-short,  brittle  iron,  which  works  easily  when 
heated  and  welds  well.  Cracks  on  the  edge  of  a  bar  are  indi- 
cations of  hot-short  iron.  Good  iron  is  readily  heated,  is  soft 
under  the  hammer,  and  throws  out  few  sparks. 

220.  Strength  of  Rope.— The  table  on  following  page  gives 
some  idea  of  the  strength  of  ordinary  Manilla  Hope. 

It  must  be  remembered  that  these  values  are  for  new  ropes 
and  that  a  few  months'  exposure  to  the  weather  will  probably 
cause  a  decrease  in  the  strength  of  40  or  50  p.  c.  A  factor  of 
safety  of  4  or  5  is  generally  employed  to  obtain  their  safe 
working  strength. 

Ropes  made  of  good  Italian  hemp  are  considerably  stronger 
than  these. 


236 


MISCELLANEOUS. 


TABLE  OF  MANILLA  ROPE-3  STRANDS. 


SIZE  OF  ROPE. 

Breaking- 
strength  in 
Ibs. 

SIZE  OF  ROPE. 

Breaking- 
strength  in 
Ibs. 

Diam.  in 
inches. 

Circum. 
in  inches. 

Diam.  in 
inches. 

Circum. 
in  inches. 

1 

j 

1 

0.71 
1.43 

2.14 
2.86 
3.57 
4.28 
5.70 

375 
1,500 
3,380 
6,000 
9,380 
13,500 
24,000 

& 

3 
§* 

f 

6 

7.14 
8.57 
10.0 
"11.4 
12.1 
14.2 
17.1 

37,500 
54.000 
73.600 
96.000 
121.000 
150,000 
216,000 

Wire  Ropes. — The  following  table  gives  the  strength  of  iron 
and  cast-steel  wire  rope  : 

TABLE   OF  IRON  AND   CAST- STEEL  WIRE  ROPE. 


SIZE  OF  ROPE. 

BREAKING- 
STRENGTH  IN  LBS. 

SIZE  OF  ROPE. 

BREAKING- 
STRENGTH  IN  LBS. 

Diam. 
in  In. 

Circum. 
in  In. 

Iron. 

C.  Steel. 

Diam. 
in  In. 

Circum. 
in  In. 

Iron. 

C.  Steel. 

1 

H 

1 

6.960 
17,280 
32,000 
54,000 

15,000 
36,000 
66,000 
104,000 

H 

* 

2J 

I 

6| 

78.000 
108,000 
130,000 
148,000 

154.000 
212,000 
250,000 
310,000 

These  ropes  have  19  wires  to  the  strand  and  hemp  centres. 
One  fifth  of  the  above  breaking-strength  may  be  taken  as  the 
safe  working  strength. 

For  the  strength  of  Iron  Rods  see  Sec.  138. 

230.  Properties  of  the  Circle. 


Diameter  X  3.14159 

Diameter  X    .88622( 

Diameter  X    .7071 

Diameter*  X    .7854 

Radius  X  6.28318 

Circumference  X    .31831 
Circumference  =  3.5449 
Diameter  =  1.1283 

Length  of  arc 


=  circumference. 

=  side  of  an  equal  square. 

=  side  of  an  inscribed  square. 

=  area  of  circle. 

=  circumference. 

=  diameter. 


area  of  circle. 


area  of  circle. 

=  number  of  degrees  X  0.017453  radius. 
Arc  of  1°  to  rad.  1  --  0.01745329. 
Arc  of  1'  to  rad.  1  =  0.000290888. 
Arc  of  1'  to  rad.  1  -  0.000004848. 
Degrees  in  arc  whose  length  =  radius  =  57°.  2957795. 
TT  =  3.1415926536;        Log  n  =  0.4971499. 


MISCELLANEOUS. 


237 


231.  PLANE  TRIGONOMETRY.-In    Fig.   96, 
angle  OAE  =  90°;  then  in  the  right-  Q  H 

angled  triangle  ABC,  if  AB  =  Radius 
=  unity, 

AF=  cosec  A; 

CE  =  versin  A; 
BH  =  co-versin  A; 
BD  =  exsec  A; 


if   the 


BC=sin  A; 
AC  =  cos  A; 
DE-ismA; 
AD  =  sec  A-, 
GF=cotA; 

Therefore 


BF  =  co-exsec  A. 


BC 


A 
cosec  A  =  - 


Thus, 
sin  A  = 


cosec  A ' 


AC 
COsA=AB> 


AB 


cos  A  = 


AC 


sec 


tan  A  = 


1 


cot  A' 


An  angle  and  its  Supplement  have  the  same  Sine  and  Cose- 
cant; but  the  Tangents,  Secants,  Cosines  and  Cotangents,  though 
of  equal  length,  are  of  contrary  signs:  so  that  in  applying  to 
obtuse  angles  trigonometrical  formulae  which  were  originally 
intended  for  acute  angles,  the  algebraic  signs  of  the  tangents, 
secants,  cosines,  and  cotangents  must  be  reversed. 

The  sine,  secant,  and  tangent  of  an  angle  A  are  respectively 
equal  to  the  cosine,  cosecant,  and  cotangent  of  its  comple- 
ment (i.e.,  of  90°  —  A). 


B  =  90°  -  A. 
AC.&C 


AB2  =  AC*  -f 

Area  of  triangle  =• 


2 


Examples  of  Right-angled  Triangles: 

1.  Given  A  =  30Q,  and  AC  -  100,  find  BC. 

T>r< 

We  see  above  that  tan  A  =  —r^, ;  therefore 


BC  =  AC  tan  A  =  57.73. 


238 


MISCELLANEOUS. 


2.  Find  the  sine  of  128°. 
Since  sin  (180°  -  A)  =  sin  A, 

sin  128°  =  sin  (180°  -  52°)  =  sin  52°, 
which  from  the  tables  we  find  =  0.788. 

Solution  of  Oblique-angled  Triangles. 


FIG.  98. 


sm  A  _  sin  B  _  sin  C 
a  b 


tan 


A-B 


2  a-+-b 


tan 


c 
A-\-B 


A    •*"*•     I     •"•     |     "-*• 

-A-  — 3—  +— g- 


5  = 


(3) 


Let 


c  =  («  +  &)_ 


A-B 


=  <;  then 


vers  ud  = 


cos 


-  ft)  (a  —  c) 


(5) 


(6) 


A 


—  a) 


(7) 


MISCELLANEOUS.  239 


Area  of  triangle  =  4/  s(s  -  a)  (s-b)  (s  -c).     .    ,    (8) 
=  ^  sin  C.      .......    (9) 

a 

a?  sin  B  sin  C 

~  ......  ° 


(11) 


The  above  formulae  are  all  that  are  required  for  the  ordi- 
nary solution  of  plane  triangles. 

Remarks.  —  Though  such  a  formula  as  No.  2  simpty  men- 
tions A  and  B  and  their  opposite  sides,  it  holds  equally  well 
whether  we  substitute  C  for  A,  or  G  for  B,  provided  that  the 
sides  are  changed  to  correspond  also.  In  Equations  2,  3,  4,  and 
5,  A  is  intended  to  represent  the  greater  angle  of  the  two 
angles  A  and  B. 

Examples,  — 

1.  Given  A,  B,  and  b,  find  A. 
By  Equation  1, 

_  b  sin  A 
~  sin  B  ' 

2.  Given  B,  c,  and  b,  find  C. 
By  Equation  1, 

~     c  sin  B 

sin  C  =  —  j  —  . 
b 

3.  Given  A,  B,  and  c,  find  a. 
By  Equation  11, 

C=  180°  -(A  +J5); 
and  by  Eq.  1, 

c  sin  A 


4.  Given  B,  a,  and  c,  find  A  and  b. 
By  Eq.  2, 

A-  C      a-c 


240  MISCELLANEOUS. 

from  which  we  obtain  the  value  of 

A-0 

2      ' 
and  by  Eq.  11, 


therefore  we  can  find  A  from  Eq.  3. 
Then  by  Eq.  5, 


__  -0. 

cos  —  -  — 


5.  Given  a,  b,  and  c,  find  B. 

By  Eq.  6, 

2(8  —  a)  (s  —  c) 


or,  we  might  equally  well  have  used  Eq.  7. 


232.  The  following  general  equations  are  worth  noting: 


sin  A  =  tan  A  cos  A  =  4/  1  —  cos2  A  =  2  sin  —  cos  —  ; 

a  4i 


cos  A  =  cot  A  sin  A  —  4/1  —  sin2  A  =  2  cos2  —  —  1; 

» 

vers  2^1  ,  .4 

tan  J.  =  sin  J.  sec  J.  =  -j — ;    (—  =  exsec  A  cot  — ; 
sm  2J.  ^ 

^ 

cot  A  —  cos  J.  cosec  A  =  — — - —r-  — 


vers  A  =  1  —  cos  A  =  2  sin2  —  =  cos  J.  exsec  A\ 

A       vers  J. 
exsec  A  =  sec  .4  -  1  =  tan  A  tan  -  = 


233. 


MISCELLANEOUS. 

Spherical  Trigonometry. 


241 


FIG.  99. 

RIGHT-ANGLED  TRIANGLES. — In  Fig.  99  let  A  =  90°;  then 
sin  b  =  sin  a  sin  B ;         tan  c  =  tan  a  cos  B; 


cot  G  =  cos  a  tan 


tan  c  =  sin  5  tan  (7; 


cos  a  =  cos  b  cose;        cos  5  =  cos  b  sin  (7; 


tan  b  tan  6 

tan  #  =  ---  -7=;        sin  c  =  —  —  —  ; 
cos  C  tan  B 


sin  b 
sin  a  =  — 

sin  5 


„      cos  J9  cos  a  sin  & 

sin  C7  = r  ;         cos  c  =  —   —  ;         sin  B  =  — 

cos  b  cos  6  sin  a 


~      tan  b  „      tan  c  _      tan  b 

cos  (7  =  —   —  ;         tan  C  =  — — =• ;         tan  B  =  —. — ; 

tan  a  sin  b  sin  c 


cos  c  = 


cos  C 
-  —  ~  ; 
sin  B 


cos  b  = 


B 


.      , 
sin  C 


cos  a  = 


cot  (7 
-  =. 
tan  B 


b  and  c  are  of  the  same  species  respectively  as  B  and  C. 

Any  side  is  greater  than  90°  if  the  other  sides  are  of  differ- 
ent species,  and  less  than  90°  if  of  the  same  species. 

B  or  (7  is  less  than  90°  if  the  containing  sides  are  of  the  same 
species,  and  less  than  90°  if  of  different  species. 


242  MISCELLANEOUS. 

Oblique-angled  triangles. 


FIG.  100. 

Let  ABC  in  Fig.  100  represent  any  oblique-angled  spherical 
triangle;  then 


sin  A  _  sin  B  _  sin  G 

sin  a  ~  sin  b  ~  sin  c  ' 


a  ~  o 

__ 


__ 


sm  - 

c  2 


tan 


tan 


f  (7   °      ~2~ 

=  cot  —  --  — 


cos 


.    a 


=  cot  -^ 


2'    .    a-f$' 

sm  ~ — 


cos  c  :=  cos  a  cos  6  -f-  sin  a  sin  5  cos  (7 ; 


sin 


A        /sin  (a-  6)  si: 

2  ~  r  sin  6  si 


—  ft)  sin  (a  —  c) 
sin  6  sin  c 


F cos  8  cos  (8  —  A\ 


'B  sin  C     ' 


(1) 


(8fl) 

(35) 

(4) 
(5) 

(6) 


a____c 
where*  =  -          —  and  8  = 


MISCELLANEOUS. 


243 


The  greater  angle  is  always  opposite  the  greater  side. 
No  angle  or  side  is  greater  than  180°. 
The  sum  of  any  two  sides  is  greater  than  the  third  side. 
The  sum  of  the  three  sides  is  less  than  360°. 
Given  a,  b,  and  C,  to  find  A  and  B :  use  Eqs.  2a  and  2b. 


"  A,  B,  and  c,  "  a  and  b 

"  a,  b,  and  G,  "  '  c; 

or,  given  a,  b,  and  C,  "  c ; 

"  A,  B,  and  a,  "  B  or  6  ; 

"  A,B,  and  a,  "  (7; 

"  A,  B,  and «,  "  c; 

"  a,  6,  and  A,  "  C; 

"  «,  b,  and  ^4,  "  c ; 

"  -4, 5,  and_c,  "  (7; 


3a  and  36. 
2a,  25,  and  36. 

4 

1 

1  and  2a. 
1  and  3a. 
1  and  2a. 
1  and  3a. 
8a,  36,  and  26. 


234.  Measures  of  length  and  surface. 

MEASURE  OF  LENGTH. 


Miles. 

Furlongs. 

Chains. 

Rods. 

Yards. 

Feet. 

Inches- 

1 

8 

80 

320 

1760 

5280 

63360 

0.125 

1 

10 

40 

220    ' 

660 

7920 

0.0125 

0.1 

1 

4 

22 

66 

792 

0.003125 

0.025 

0.25 

1 

55 

16.5 

198 

0.00056818 

0.0045454 

0.045454 

0.181818 

1 

3 

36 

0.0(1018939 

0.00151515 

0.01515151 

0.0600060 

0.33333 

1 

12 

0.000015783 

0.000126262 

0.001262626 

0.00505050 

0.0277777 

0.083333 

1 

MEASURE  OF  SURFACE. 


Sq.  Miles. 

Acres. 

S.  Chains. 

Sq.  Rods. 

Sq.  Yards. 

Sq.  Feet. 

1 

640 

6400 

102400 

3097600 

27878400 

0.001562 

1 

10 

160 

4840 

43560 

0.0001562 

0.1 

1 

16 

484 

4356 

0.000009764 

0.00625 

0  0625 

1 

30.25 

272.25 

0.000000323 

0.0002066 

0.002066 

0.0330 

1 

9 

0.0000000358 

0.00002296 

0.0002296 

0.00367 

0.1111111 

1 

244 


MISCELLANEOUS. 


235.  Measures  of  weight  and  capacity. 
MEASURES  OF  WEIGHTS. 

AYOIRDUPOIS. 


Ton. 

Cvt. 

Pounds. 

Ounces. 

Drams. 

1 

20 

2240 

35840 

573440 

0.05 

1 

112 

1792 

28672 

0.00044642 

0.0089285 

1 

16 

256 

0.00002790 

0.000558 

0.0625 

1 

16 

0.00000174 

0.0000348 

0.0016 

0.0625 

1 

TROY. 


Pounds. 

Ounces. 

Dwt. 

Grains. 

Pound  Avoir. 

1 

12 

240 

5760 

0.822861 

0.08:3333 

1 

20 

480 

0.068571 

0.004166 

0.05000 

1 

24 

0.0034285 

0.0001736 

0.002083333 

0.0416666 

1 

0.00014285 

1.215275 

14.58333 

291.6666 

7000 

1 

MEASURE  OF  CAPACITY, 


Cub.  Yard. 

Bushel. 

Cub.  Feet. 

Pecks. 

Gallons. 

Cub.  inch. 

1 

0.03961 
0.037037 
0.009259 

21.6962 
1 
0.803564 
0.25 
0.107421 

27 
1.24445 
1 

0.31114 
0.133681 
0.000547 

100.987 
4 
3.21425 
1 
0.429684 
0.001860 

201.974 
9.30918 
7.4805 
2.32729 
1 
0.004329 

46656 
2150.42 
1728 
537.605 
231 
1 

APPENDIX. 


NOTE  A.    (See  Sec.  10.) 

IF  we  knew  the  average  pressure  in  the  cylinders  we  could 
find  the  propelling  force  of  an  engine  at  any  speed,  if  not 
limited  by  adhesion,  by  the  following  rule  : 

Multiply  together  the  square  of  the  diameter  of  one  piston 
in  inches,  the  length  of  stroke  in  inches,  and  the  mean 
pressure  (above  atmosphere)  in  Ibs.  per  sq.  in.  The  product 
divided  by  the  diameter  of  a  driver  in  inches  gives  the  pro- 
pelling force  in  Ibs.,  ignoring  "  internal  frictional  resistances." 

Theoretically,  the  mean  effective  cylinder- pressure  in  Ibs. 
per  sq.  in.  equals 

P  +  2.3P(Log£) 


8 


-15, 


where  P  =  absolute  boiler-pressure  in  Ibs.  per  sq.  in.  and  8  = 
Stroke  -*-  part  of  stroke  before  cut-off. 

But  owing  to  the  contraction  of  the  steam-ports,  the  initial 
cylinder-pressure  always  falls  below  the  boiler- pressure. 
Similarly  owing  to  the  contraction  of  the  exhaust-port,  back- 
pressure always  exists ;  and  these  are  matters  so  purely  of 
mechanical  detail  that  no  general  rule  can  be  given  which 
would  take  them  into  consideration. 

At  20  miles  per  hour,  however,  the  effective  initial  cylinder 
pressure  often  equals  only  about  90  p.  c.  of  the  boiler-pres- 
sure, and  at  50  m.  p.  h.  about  60  p.  c. 

Thus  if  P  =  125  Ibs.  per  sq.  in.  and  the'stroke  =  24  inches  ; 
if  steam  is  cut  off  at  6  inches,  the  theoretical  mean  cylinder- 
pressure  =  59  Ibs.  per  square  inch,  which  at  50  m.  p.  h.  will 
probably  be  reduced  to  about  36  Ibs. :  so  that  if  the  diameter 
of  the  piston  =16  inches,  and  of  the  driving-wheels  60 
inches,  the  propelling  force  will  equal  3680  Ibs.;  and  if  we 
deduct  10  p.  c.  from  this  for  internal  frictioual  resistances,  the 
propelling  force  =  3200  Ibs. 

245 


246  APPENDIX. 

NOTE  B.    (See  Sec.  19.) 

In  order  to  reduce  the  quantities  used  in  Diagram  II  into 
the  same  units,  say  ton,  mile,  and  hour,  the  ordiuates  of  the 
curves  must  be  multiplied  by 


(3600)2 

aooo-xi»ox88-8=40(Iie8rly) 

to  reduce  them  to  tons  weight  (2000  Ibs.),  in  miles  per  hour 
units.     Then,  with  the  units  selected,  the  equation  of  motion  is 


But  if  x  is  the  space  passed  over, 


so  that 

and  therefore 


OQ.d(OQ) 
~  = 


the  graphic  process  giving  the  integral.  But  with  the 
scales  used  in  Diagram  II,  instead  of  multiplying  the  ordinatea 
as  above,  we  can  simply  use  as  a  scale  1  square  inch  =  1  mile, 
which  practically  comes  to  the  same  thing.  If  the  horizontal 
scale  were  ten  miles  per  hour  to  one  inch,  the  scale  then  to  be 
used  would  be  4  square  inches  =  1  mile;  and  this  is  often  a 
more  convenient  scale  to  adopt. 

NOTE  C.     (See  Sec.  44.) 

Messrs.  W.  and  L.  E.  Gurley  in  their  Manual  give  the  fol- 
lowing methods  of  adjusting  the  object-slide  : 

To  Adjust  the  Object-slide  of  a  Transit.—"  Hav- 

ing set  up  and  levelled  the  instrument,  the  line  of  collimation 
being  also  adjusted  for  objects  from  three  hundred  to  five 


APPENDIX.  247 

hundred  feet  distant,  clamp  the  plates  securely,  and  fix  the 
vertical  cross-wire  upon  an  object  as  distant  as  may  be  dis- 
tinctly seen ;  then,  without  disturbing  the  instrument,  throw 
out  the  object-glass,  so  as  to  bring  the  vertical  wire  upon  an 
object  as  near  as  the  range  of  the  telescope  will  allow.  Hav- 
ing this  clearly  in  mind,  unclamp  the  limb,  turn  the  instru- 
ment half-way  around,  reverse  the  eye-end  of  the  telescope, 
clamp  the  limb,  and  with  the  tangent-screw  bring  the  vertical 
wire  again  upon  the  near  object ;  then  draw  in  the  object-glass 
slide  until  the  distant  object  first  sighted  upon  is  brought  into 
distinct  vision.  If  the  vertical  wire  strikes  the  same  line  as  at 
first,  the  slide  is  correct  for  both  near  and  remote  objects ; 
and,  being  itself  straight,  for  all  distances. 

"  But  if  there  be  an  error,  proceed  as  follows:  First,  with  the 
thumb  and  forefinger  twist  off  the  thin  brass  tube  that  covers 
the  screws.  Next,  with  the  screw-driver,  turn  the  two  screws 
on  the  opposite  sides  of  the  telescope,  loosening  one  and 
tightening  the  other,  so  as  apparently  to  increase  the  error, 
making,  by  estimation,  one-half  the  correction  required. 

"Then  go  over  the  usual  adjustment  of  the  line  of  collima- 
tion,  and  having  it  completed,  repeat  the  operation  above  de- 
scribed ;  first  sighting  upon  the  distant  object,  then  finding  a 
near  one  in  line,  and  then  reversing,  making  correction,  etc., 
until  the  adjustment  is  complete." 

To  Adjust  the  Object-slide  of  a  Y-Level.—"  The 
maker  selects  an  object  as  distant  as  may  be  distinctly  ob- 
served, and  upon  it  adjusts  the  line  of  collimation,  making 
the  centre  of  the  wires  to  revolve  without  passing  either  above 
or  below  the  point  or  line  assumed. 

"In  this  position,  the  slide  will  be  drawn  in  nearly  as  far  as 
the  telescope-tube  will  allow. 

"  He  then,  with  the  pinion-head,  moves  out  the  slide  until  an 
object,  distant  about  ten  or  fifteen  feet,  is,  brought  clearly  into 
view  ;  again  revolving  the  telescope  in  the  Y's,  he  observes 
whether  the  wires  will  reverse  upon  this  second  object. 

"  Should  this  happen  to  be  the  case,  he  will  assume  that,  as 
the  line  of  collimation  is  in  adjustment  for  these  two  dis- 
tances, it  will  be  so  for  all  intermediate  ones,  since  the  bear- 
ings of  the  slide  are  supposed  to  be  true,  and  their  planes 
parallel  with  each  other. 

"If,  however,  as  is  most  probable,  either  or  both  wires  fail  to 


248  APPENDIX. 

reverse  upon  the  second  point,  he  must  then,  by  estimation, 
remove  half  the  error  by  the  screws  at  right  angles  to  the 
hair  sought  to  be  corrected,  remembering,  at  the  same  time, 
that  on  account  of  the  inverting  property  of  the  eye -piece  he 
must  move  the  slide  in  the  direction  which  apparently  in- 
creases the  error.  When  both  wires  have  thus  been  treated 
in  succession,  the  line  of  collimatiou  is  adjusted  on  the  near 
object,  and  the  telescope  again  brought  upon  the  most  distant 
point ;  here  the  tube  is  again  revolved,  the  reversion  of  the 
wires  upon  the  object  once  more  tested,  and  the  correction,  if 
necessary,  made  in  precisely  the  same  manner. 

"  He  proceeds  thus,  until  the  wires  will  reverse  upon  both 
objects  in  succession  ;  the  line  of  collimation  will  then  be  in 
adjustment  at  these  and  all  intermediate  points,  and  by  bring- 
ing the  screw-heads,  in  the  course  of  the  operation,  to  a  firm 
bearing  upon  the  washers  beneath  them,  the  adjustable  ring 
will  be  fastened  so  as  for  many  years  to  need  no  further  ad- 
justment." 

"  The  centring  of  the  eye-tube  is  performed  after  the  wires 
have  been  adjusted,  and  is  effected  by  moving  the  ring,  by 
means  of  the  screws  shown  on  the  outside  of  the  tube,  until 
the  intersection  of  the  wires  is  brought  into  the  centre  of  the 
field  of  view." 

NOTE  D.     (See  Sec.  57.) 

The  time  at  which  any  elongation  will  occur  may  be  found 
by  the  formula 

cos  Ji  —  cot  (dec.)  X  tan  (lat.), 

where  Ti  —  the  hour-angle  (see  Sec.  182),  h  really  being  the 
supplement  of  the  angle  at  P  in  the  right-angled  spherical 
triangle  WZP  (or  EZP]  in  Fig.  10,  the  right  angle  beinir  :lt 
WorE. 

The  angle  li  may  be  reduced  to  mean  time  as  shown  in  Part 
III. 

NOTE  E.     (See  Sec.  57.) 

To  find  the  azimuth  of  two  stars  when  in  the  same  vertical 
plane  (Polaris  being  one  of  them)  proceed  as  follows  : 

Let  A  =  the  difference  in  R.A.  of  the  stars, 

d  =  the  declination  of  Polaris, 
and  D  —  the  declination  of  the  other  star. 


APPENDIX.  249 

Find  p  and  m  from  the  formulae 

cos  A  sin  D 

tan  m  —  —    --,        p  =  ---  ; 
tan  D  cos  m 

then  find  a  from  the  formula 

cos  a  =  p  sin  (d  +  w). 
Then  Z,  the  azimuth,  is  given  by 

sin  A  cos  D  cos  d 
sin  Z  =  -  -  —  .  -  , 
cos  L  sin  a 

where  L  —  the  latitude  of  the  place. 

To  find  the  interval  of  time  which  must  elapse  after  the  two 
stars  are  observed  to  be  in  the  same  vertical  plane,  before 
Polaris  will  be  due  north,  find  S  from  the  equation 

.  cos  D 

sin  8=  sin  A  —  -  . 
sm  a 

Then 

L4-d 


where  h  is  the  hour-angle  in  sidereal  time.  . 

To  find  the  interval  in  mean  time,  see  Sec.  179. 

The  above  steps  may  be  easily  traced  by  drawing  the  posi- 
tions of  the  star,  the  pole,  and  the  zenith. 

It  is  not  necessary  to  use  Polaris  ;  but  if  any  other  star  is 
selected,  d  refers  to  the  star  whose  declination  is  the  greater. 

NOTE  F.     (See  Sec.  58.) 
The  true  value  of  the  convergence  is  given  by  the  equation 

.    convergence        .    diff.  of  long. 
sin  —  —  =  sm  —  2-  x  sin  (lat.). 

a  .  4/ 

If  the  places  are  in  different  latitudes,  as  A  and  D  in  Fig. 
12,  we  have  the  convergence  =  the  difference  in  azimuth  at 


250  APPENDIX. 

A  and  D,  which  we  can  find  by  solving  the  spherical  triangle 
AND. 

NOTE  Gr.    (See  Sec.  189.) 

The  difference  in  altitude  in  seconds  of  arc,  between  the 
meridian  altitude  and  the  maximum  altitude  of  a  body,  is 
equal  to 

d2 


where 


_  cos  lat.  cos  dec.  X  1.964 

sin  (lat.  —  dec.)        ' 


and  d  =  the  hourly  change  of  declination  in  minutes  of  arc. 

When  the  declination  differs  in  sign  from  the  latitude,  it 
will  be  negative.  If  the  body  has  its  declination  changing 
towards  the  north  in  the  northern  hemisphere  or  towards  the 
south  in  the  southern  hemisphere,  the  meridian  altitude  pre- 
cedes the  maximum  altitude,  which  will  be  the  case  between 
mid-winter  and  mid-summer ;  but  if  changing  towards  the 
south  in  the  northern  hemisphere,  or  towards  the  north  in  the 
southern,  the  maximum  altitude  occurs  to  the  east  of  the 
meridian. 


TABLES. 


TABLE  I.-RADII. 


Deg. 

Radius. 

Deg. 

Radius. 

Deg. 

Radius. 

Deg. 

Radius. 

Deg. 

Radius. 

o°  o' 

Infinite 

1°  0' 

5729.65 

2°  0' 

2864.93 

3°  0' 

1910.08 

4°  0' 

1432.69 

i 

343775. 

1 

5635.72 

1 

2841.26  "   1 

1899.53 

1 

1426^74 

2 

171887. 

2 

5544.83 

2 

2817.97 

2 

1889  09 

2 

1420  85 

3 

114592. 

3 

5456.82 

3 

2795.06 

3 

187S.77 

3 

1415.01 

4 

85943.7 

4 

5371.56 

4 

2772.53 

4 

1868.56 

4 

1409.21 

5 

68?54.9 

5 

5288.92 

5 

2750.35 

5 

1858.47 

5 

1403  46 

6 

57295.8 

6 

5208.79 

6 

2728  52 

6 

1848.48 

6 

1397^76 

7 

49110.7 

7 

5131.05 

7 

2707.04 

1838  59 

1392  10 

8 

42971.8 

8 

5055.59 

8 

2685.89 

8 

1828.82 

8 

1386.49 

9 

38197.2 

9 

4982.33 

9 

2665.08 

9 

1819.14 

9 

1380.92 

10 

34377.5 

10 

4911.15 

10 

2644.58 

10 

1809.57 

10 

1375.40 

11 

31252.3 

11 

4841.98 

11 

2624.39 

11 

1800.10 

11 

1369.92 

12 

28647.8 

12 

4774.74 

12 

2604.51 

12 

1790.73 

12 

1364.49 

13 

26444.2 

13 

4709.33 

18 

2584.93 

13 

1781.45 

13 

1359.10 

14 

24555.4 

14 

4645.69 

14 

2565.65 

14 

1772  27 

14 

1353.75 

15 

22918.3 

15 

4583.75 

15 

2546.64 

15 

1763!  18 

15 

1348.45 

16 

21485.9 

16 

4523.44 

16 

2527.92 

16 

1754.19 

16 

1343.15 

17 

20222.1 

17 

4464.  70 

17 

2509.47 

17 

1745  26 

17 

1337  65 

18 

19098.6 

18 

4407.46 

18 

2491.29 

18 

1736.48 

18 

1332.77 

19 

18093.4 

19 

4351.67 

19 

2473.37 

19 

1727.75 

19 

1327  63 

20 

17188.8 

20 

4297.28 

20 

2455.70 

20 

1719.12 

20 

1322^53 

21 

16370  2 

21 

4244.23 

21 

2438.29 

21 

1710  56 

21 

1317.46 

22 

15626.1 

22 

4192.47 

22 

2421.12 

22 

1702.10 

22 

1312.43 

23 

14946.7 

23 

4141.96 

23 

2404.19 

23 

1093.72 

23 

1307.45 

24 

14323.6 

24 

4092.66' 

24 

2387.50 

24 

1685.42 

24 

1302.50 

25 

13751.0; 

25 

4044  51  1 

25 

2371.04 

25 

1677.20 

25 

1297.58 

26 

13222.1 

26 

3997.49 

26 

2354.80 

26 

1669.06 

26 

1292.71 

27 

12732.41 

27 

3951.54 

27 

2338.78 

1661.00 

27 

1287.87 

28 

12277.7 

28 

3906.54 

28 

2322.98 

28 

1653.01 

28 

1283  07 

29 

11851.3 

29 

3862.74 

29 

2307.39 

29 

1645.11 

29 

1278  30 

30 

11459.2 

30 

3819.83 

30 

2292.01 

30 

1637.28 

30 

1273.57 

31 

11089  6 

31 

3777.85 

31 

2276.84 

31 

1629.52 

31 

1268.87 

32 

10743  Oi 

32 

3736.79 

32 

2261.86 

32 

1621.84 

32 

1204.21 

33 

10417.5 

88 

3696  61 

33 

2247.08 

33 

1614.22 

33 

1  259  .  5S 

34 

10111.1 

34 

3657.29 

34 

2232  49 

34 

1606.68 

34 

1254^98 

35 

9822.18 

35 

3618.80! 

35 

2218^09 

35 

1599.21 

35 

1250.42 

36 

9549.34 

36 

3581.10; 

36  2203.87 

36 

1591.81 

36 

1245.89 

37 

9291.29 

37 

3544.19 

37 

2189.84 

37 

1584.48 

37 

1241.40 

38 

9046.75 

38 

3508.  02  ! 

38 

2175.  98  | 

38 

1577.21 

38 

1236.94 

39 
40 

8814.78 
8594.42 

39 
40 

3472.59 
3437.87 

39 
40 

2162.30! 
2148.79 

39 

40 

1570.01 
1562.88 

39 
40 

1232.51 
1228.11 

41 

8384.80 

41 

3403.83 

41 

2135.44 

41 

1555.81 

41 

1223  74 

44 

8185.16 

42 

3370.46! 

42 

2122.26 

42 

1548.80 

42 

1219.40 

43 

7994.81 

43 

3337.  74  i 

43 

2109.24 

43 

1541.  SO 

43 

1215.30 

44 

7813.11 

44 

3305.65 

44  2096.39 

44 

15:34.98 

44 

1210.82 

45 

7639.49 

-  45 

3274.17 

45  2083.68 

45 

1528.16 

45 

1206.57 

46 

7473.42 

46 

3243.29 

40  2071.13 

46 

1521.40 

46 

1202.36 

47 

7314.41 

47 

3212.98 

47 

2058.73! 

47 

1514.70 

47 

1198.17 

48 

7162.03 

48 

3183.23 

48 

2046.48] 

48 

1508.06 

48 

1194.01 

49 

7015.87 

49 

3154.03 

49 

2034.37 

49 

1501.48 

49 

1189.88 

50 

6875.55 

50 

3125.36 

50 

2022.41 

50 

1494.95 

50 

1185.78 

51 

6740.74 

51 

3097.20 

51 

2010.59; 

51 

1488.48 

51 

1181.71 

52 

6611.12 

52 

3069.551 

52 

1998.90 

52 

1482.07 

52 

1177.66 

53 

6486.38 

53 

3042.39 

58 

1987.  35  1 

53 

1475.71- 

53. 

1173.65 

54 

6366.26 

54 

3015.71 

54 

1975.93 

54 

1101).  11 

54 

1169.66 

55 

6250.51 

55 

2989.48 

55 

1964.64 

55 

1463.16 

55 

1105.70 

56 

6138.90 

56 

2963.  71  ! 

56  1953.48 

56 

1456.96 

56 

1161.76 

57 

6031.20 

57  2938.  39  i 

57 

1942.44! 

57 

1450.81 

57 

1157.85 

58 

5927.22 

58 

2913.49 

58 

1931.53 

58 

1444.72 

58 

1153.97 

59 

5826.76! 

59 

2889.01 

59 

1920.75 

59 

1438.68 

59 

1150.11 

60 

5729.65 

60 

2864  ..93 

60 

1910.08 

60 

1432.69 

60 

1146.28 

252 


TABLE  I. -RADII. 


Deg. 

adius. 

eg.  [  Radius. 

eg.  1 

Radius. 

eg. 

adius.  | 

eg. 

adius. 

5°  0' 

1-16.28 

0'  955.366 

0'  819.020) 

0' 

16.7791. 

0' 

37.275 

1 

142.47 

1  952.722 

1  817.  0^7 

1 

15.  391  ! 

1 

36.099 

2 

138.09 

2  950.093 

2  815.144 

2 

13.8101 

2 

34.928 

3 

134.94 

3  947.478 

3  813.238; 

3 

12.335 

3 

33.761 

4 

131.21 

4  944.877 

4  i  811.303 

4 

10.8651 

4 

32.599 

5 

127.50 

5  i  942.291 

5  809.397 

5 

09.402 

5 

31.440 

6 

123.82 

6  939.719 

6  807.499 

6 

07.945 

6 

30.286 

« 

120.16 

7 

937.161 

7 

805.611 

7 

06.493! 

7 

29.136 

8 

116.52 

g 

934.  GIG 

8 

803.731 

8 

05.048 

8 

27.991 

9 

112  91 

9 

932.086 

9 

801.860 

9 

"•03.609 

9 

626.849 

10 

109.33 

10 

929.569 

10 

799.997 

10 

"02.175 

10 

625.712 

11 

105.76 

11 

927.066 

11 

798.144 

11 

-00.748 

11 

624.579 

12 

102.2-.' 

12 

924  576 

12 

796.299 

12 

699.326 

12 

623.450  ; 

13 

098.70 

13 

922.100 

13 

794.462 

13 

697.910 

13 

622.325 

14 

095.20 

14 

919.637 

14 

792.634 

14 

696.499 

14 

621.203 

15 

091.73 

15 

917.187 

15 

790.814 

15 

195.095 

15 

620  087 

16 

088.28 

16 

914.750 

16 

789.003 

16 

693.  696  > 

16 

618.974 

17 

1084.85 

912  326 

17 

787.210 

17 

692.302 

17 

617.865 

18 

1081.44 

18 

909.915 

18 

785.405 

18 

600.914 

18 

616.760 

19 

1078.05 

19 

907.517 

19 

783.618 

19 

689.532 

19 

615.660 

20 

1074.68 

20 

905.131 

20 

781.840 

20 

688.156 

20 

614.563 

21 

1071.34 

21 

902.758 

21 

780.069 

21 

686.785 

21 

613.470 

22 

1068.01 

22 

900.397 

22 

778.307 

22 

685.419 

22 

612.380 

23 

1064.71 

23 

898.048 

23 

776.552 

23 

684.059 

23 

611.295 

24 

1061.43 

24 

895.712 

24 

774.806 

24 

682.704! 

24 

610.214 

25 

1058.16 

25 

893.3HS 

25 

773.067 

25 

681.354 

25 

609.136 

26 

1054.92 

26 

891.076 

26 

771.336 

26 

680.010 

26 

608.062 

27 

1051.70 

27 

888.776 

27 

769.613 

27 

678.671 

27 

606.992 

28 

1048.48 

28 

886.488 

28 

767.897 

28 

677.338 

28 

605.926 

29 

1045.31 

29 

884.211 

29  766.190 

29 

676.008 

29 

604.864 

30 

1042.14 

30 

881.946 

30 

764.489 

30 

674.686 

30 

603.805 

31 

1039.00 

31 

879.693 

31 

762.797 

31 

673.369 

31 

602.750 

32 

1035.87 

32 

877.451 

32 

761.112 

32 

672.056 

32 

601.698 

33 

1032.76 

33 

875.221 

33 

759.434 

33 

670.748 

33 

600.651 

34 

1029.67 

34 

873.002 

34 

757.764 

34 

669.446 

34 

599.607 

35 

1026.60 

35 

870.795 

35 

756.101 

35 

668.148 

85 

598.567 

36 

1023.55 

36 

868.598 

36 

754.445 

36 

666.856 

36 

597.530 

37 

1020.51 

37 

866.412 

37 

752.796 

37 

665.568 

37 

596.497 

38 

1017.49 

38 

864.238 

38 

751.155 

38 

-664.286 

38 

595.467 

39 

1014.50 

39 

862.075 

39 

749.521 

39 

663.008 

39 

594.441 

40 

1011.51 

40 

859.922 

30 

747.894 

40 

661.736 

40 

593.419 

41 

1008.5. 

41 

857.780 

41 

746.274 

41 

660.468 

41 

592.400 

43 

1005  ft 

42  855.648 

42 

744.661 

42 

659.205 

42 

591.384 

43 

1002.6 

43  853.527 

43 

743.055 

4: 

657.947 

43 

590.372 

44 

999.76 

44 

851.417 

44 

741.45 

44 

656.694 

44 

589.364 

45 

996.86 

45 

849.31 

45 

739.86 

45 

655.446 

45 

588.359 

46 

993.98 

46 

847.22 

46 

738.27 

46 

654.202 

46 

587.357 

47 

991.12 

47 

845.14 

47 

736.70 

4" 

652.963 

47 

586.359 

48 

988.28 

48 

843.08 

48 

735.12 

48 

651.729 

48 

585.364 

49 

985.45 

49 

841.02 

49 

733.56 

49 

650.499 

49 

584.373 

50 

982.638 

50 

838.97 

50 

732.00 

50 

649.274 

50 

583.385 

5 

979.84 

51 

836.93 

51 

730.45 

5 

648.05 

5 

582.400 

5 

977.06 

52 

834.90 

52 

728.90 

5 

646.83 

5 

581.419 

53 

974.29 

53 

832.88 

53 

7^7  .3  i 

5 

645.62 

5 

580  441 

5 

971.54 

54 

830.8" 

54 

725.83 

5 

644.42 

5 

579.466 

5 

968.81 

55 

828.8~ 

55  724.31 

5 

643.21 

578.494 

5 

966.00 

56 

826.88 

56  722.79 

5 

642.02 

577.526 

5' 

963.38 

57 

824.90 

57  721.28 

5' 

640.82 

576.561 

5 

960.69 

58 

822.9 

58  719.77 

5 

639.63 

575.599 

5 

958.02 

59 

820.9 

59  718.27 

5 

638.45 

574.641 

6 

955.36 

60 

819.0 

60  716.77 

60 

637.27 

( 

573.686 

TABLE  I.-RADII. 


Deg. 

Radius. 

Deg. 

Radius. 

Deg. 

Radius. 

Deg. 

Radius. 

Deg. 

Radius. 

10°  0' 

573.  C86 

12°  0' 

478.  a39 

14°  0' 

410.275 

16°  0' 

359.265 

18°  0' 

319.623 

2 

4 

571.784: 
569.896 

2 

4 

477.018 
475.705 

2 
4 

409.306 
40H.341 

0 

4 

358.523 

357.784 

2 

4 

319  037 
318.463 

6 

568.  (KO 

6 

474.400 

6 

407.380 

6 

357.048 

6 

317.871 

8 

566.156 

8 

473.102 

8 

406.424 

8 

356.315 

8 

317.292 

10 

564.305 

10 

471.810 

10 

405.473 

10 

355.585 

10 

316.715 

12 

562.466 

12 

470.526 

12 

404.526 

12 

354.859 

12 

316.139 

14 

560.638 

14 

469  249 

14 

403.583 

14 

354.135 

14 

315.566 

16 

558.823 

16 

467.978 

16 

402.645 

16 

353.414 

16 

314.993 

18 

557.019 

18 

466.715 

18 

401  .  712 

18 

352.696 

18 

314.426 

20 

555.227 

20 

465.459 

20 

400.782 

20 

351.981 

20 

313.860 

22 

553.447 

22 

464.209 

22 

399.857 

22 

351.269 

22 

313.295 

24 

551.678 

24 

462.966 

24 

398.937 

24 

350  560 

24 

312.732 

26 

549.920 

26 

461  .  729 

26 

398.020 

26 

349.854 

26 

312.172 

28 

548.174 

28 

460.500 

28 

397.108 

28 

349.150 

28 

311  613 

30 

546.438 

30 

459.276 

30 

396.200 

30 

348.450 

30 

311.056 

32 

544.714 

32 

458.060 

32 

395.296 

32 

347.752 

32 

310.502 

34 

543.001 

34 

456.850 

34 

394.396 

34 

347.057 

34 

309.949 

36 

541.298 

36 

455.646 

36 

393.501 

36 

346.365 

36 

309.399 

38 

539.606 

38 

454.449 

38 

392.609; 

38 

345.676 

38 

308.850 

40 

537.924 

40 

453.259 

40 

391.722 

40 

344.990 

40 

308.303 

42 

536.253 

42 

452.073 

42 

390.  838  i 

42 

344.306 

42 

307.759 

44 

534.593 

44 

450.  894  j 

44 

389.959 

44 

343.625 

44 

807.216 

46 

532.943 

46 

449.722! 

46 

389.084: 

46 

342.947 

46 

306.675 

48 

531.303 

48 

448.556; 

48 

388.212 

48 

342.271 

48 

306.136 

50 

529.673 

50 

447.395 

50 

387.345 

50 

341.598 

50 

305.599 

52 

528.053 

52 

446.241 

52 

386.481 

52 

340.928 

52 

305.064 

54 

526  443 

54 

445.093 

54 

385.621 

54 

340.2(50 

54 

304.531 

56 

524.843 

56 

443.951 

56 

384.765 

56 

339.595 

56 

304.000 

58 

523.252 

58 

442.814 

58 

383.913 

58 

338.933 

58 

303.470 

11°  0' 

521.671 

13°  0' 

441.684 

15°  0' 

383.065 

17°  0' 

338.273 

19°  0' 

302.943 

2 

520.100 

2 

440.559: 

2 

382.220: 

2 

337.616 

2 

302.417 

4 

518.539 

4 

439  440: 

4 

381.380 

4 

336.962 

4 

301.893 

6 

516.986 

6 

488.  326  | 

6 

380.543 

6 

336.310 

6 

301.371 

8 

515.443 

8 

437.219 

8 

379.709 

8 

335.660 

8 

300.851 

10 

513.909 

10 

436.117 

10 

378.880i 

10 

335.013 

10 

300.3:33 

12 

512.385 

12 

435.020 

12 

378.054! 

12 

334.369 

12 

299.816 

14 

510.869 

14 

433.929 

14 

377.231! 

14 

333.727 

14 

299.302 

16 

509.363 

16 

432.844 

16 

376.412! 

16 

333.088 

16 

298.789 

18 

507.865 

18 

431.764 

18 

375.597 

18 

332.451 

18 

298.278 

20 

506.376 

20 

430.690 

20 

374.786 

20 

331.816 

20 

297.768 

22 

504.896 

22 

429.620 

22 

373.  977  j 

22 

331.184 

22 

297.260 

24 

503.425 

24 

428.557 

24 

373.173 

24 

330.555 

24 

296.755 

26 

501.96'J 

26 

427.498 

26 

372.372 

26 

329.928 

26 

296.250 

28 

500.507 

28 

426.145 

28 

371.574 

28 

329  303 

28 

295.748 

30 

499.061 

30 

425.396 

30 

370.780 

30 

328.689! 

30 

295.247 

32 

497.624 

32 

424.354 

32 

369.989 

32 

328.061 

32 

294.748 

34 

496.195 

34 

423.316 

34 

369.2021 

34 

327.443 

34 

294.251 

36 

494.774 

36 

422.283 

36 

368.418 

36 

326.828 

36 

293.756 

38 

493.361 

38 

421.256 

38 

367.637 

38 

326.215 

38 

293.262 

40 

491.956 

40 

420.233 

40 

366.859 

40 

325.604 

40 

292.770 

42 

490.559 

42 

419.215 

42 

»  56.  085 

42 

324.996 

42 

292.279 

44 

489.171 

44 

418.203 

44 

365.315 

44 

324  390 

44 

291.790 

46 

487.790 

46 

417.195 

46 

3(51.547 

46 

323.786 

46 

291.303 

48 

486.417 

48 

416.192 

48 

3f,3.783 

48 

323.184 

48 

290.818 

50 

485  051 

50 

415.194 

50 

363.022 

50 

3-22.  585 

50 

290.331 

52 

483.694 

52 

414.201 

52 

362.264 

52 

3-21.989 

52 

289.851 

54 

482.344 

54 

413.212 

54 

361.510 

54 

321.394 

54 

289.371 

56 

481  001 

56 

412.229 

56 

3(50.758 

56 

320.801 

56 

288.892 

58 

479.666 

58 

411.250 

58 

360.010 

58 

320.211 

58 

288.414 

60 

478.339 

60 

410.275 

60 

359.265 

60 

319.623 

60 

287.939 

254 


TABLE  II.— TANGENTS  AND  EXTERNALS  TO  A  1°  CURVE. 


Angle. 
I. 

Tan- 
gent. 

T. 

Exter- 
nal. 

E. 

Angle. 
I. 

Tan- 
gent. 

T. 

Exter- 
nal. 

E. 

Angle. 
I. 

Tan- 
gent. 

T. 

Exter- 
nal. 

E. 

1° 

50.00 

.218 

11 

551.70 

26.500 

21° 

1061.9 

97.577 

10' 

58.34 

.297 

10' 

560.11 

27.313 

10' 

1070.6 

99.155 

20 

60.67 

.388 

20 

568.53 

28.137 

20 

1079.2 

100.75 

30 

75.01 

.491 

30 

576.95 

28.974 

30 

1087.8 

102.35 

40 

83.34 

.606 

40 

585.36 

29.824 

40 

1096.4 

103.97 

50 

91.68 

.733 

50 

593.79 

30.686 

50 

1105.1 

105.00 

2 

100.01 

.873 

12 

602.21 

31.561 

22 

1113.7 

107.24 

10 

108.35 

1.024 

10 

610.64 

32.447 

10 

1122.4 

108.90 

20 

110.08 

1.188 

20 

619.07 

33.347 

20 

1131.0 

110.57 

30 

125.02 

1.364 

30 

627.50 

34.259 

30 

1139.7 

112.25 

40 

133.36 

1.552 

40 

635.93 

35.183 

40 

1148.4 

113.95 

50 

141.70 

1.752 

50 

644.37 

36.120 

50 

1157.0 

115.66 

3 

150.04 

1.964 

13 

652.81 

37.070 

23 

1165.7 

117.38 

10 

158.38 

2.188 

10 

661.25 

38.031 

10 

1174.4 

119.12 

20 

166.72 

2.425 

20 

669.70 

39.006 

20 

1183.1 

120.87 

30 

175.06 

2.674 

30 

678.15 

39.993 

30 

1191.8 

122.63 

40 

183.40 

2.934 

40 

686.6,) 

40.992 

40 

1200.5 

124.41 

50 

191.74 

3.207 

50 

695.06 

42.004 

50 

1209.2 

126.20 

4 

200.08 

3.492 

14 

703.51 

43.029 

24 

1217.9 

128.00 

10 

208.43 

3.790 

10 

711.97 

44.066 

10 

1226.6 

129.82 

20 

216.77 

4.099 

20 

720.44 

45.116 

20 

1235.3 

131.65 

30 

225.12 

4.421 

30 

728.90 

46.178 

30 

1244.0 

133.50 

40 

233.47 

4.755 

40 

737.37 

47.253 

40 

1252.8 

135.35 

50 

241.81 

5.100 

50 

745.85 

48.341 

50 

1261.5 

137.23 

5 

250.16 

5.459 

15 

754.32 

49.441 

25 

127'0.2 

139.11 

10 

258.51 

5.829 

10 

762.80 

50.554 

10 

1279.0 

141.01 

20 

266.86 

6.211 

20 

771.»9 

51.679 

20 

1287.7 

142.93 

30 

275.21 

6.606 

30 

779.77 

52.818 

30 

1290.5 

144.85 

40 

283.57 

7.013 

40 

788.26 

53.969 

40 

1305.3 

146  79 

50 

291.92 

7.432 

50 

796.75 

55.132. 

50 

1314.0 

148.75 

6 

300.28 

7.863 

16 

805.25 

56.309 

26 

1322.8 

150.71 

10 

308.64 

8.307 

10 

813.75 

57.498 

10 

1331.6 

152.69 

20 

316.99 

8.762 

20 

822.25 

58.699 

20 

1340.4 

154.09 

30 

325.35 

9.230 

30 

830.76 

59.914 

30 

1349.2 

156.70 

40 

333.71 

9.710 

40 

&S9.27 

61.141 

40 

1358.0 

158.72 

50 

342.08 

10.202 

50 

847.78 

62.381 

•       50 

1366.8 

160.76 

7 

350.44 

10.707 

17 

856.30 

63.634 

27 

1375.6 

162.81 

10 

358.81 

11.224 

10 

864.82 

64.900 

10 

1384.4 

164.86 

20 

367.17 

11.753 

20 

873.35 

66.178 

20 

1393.2 

166.95 

30 

375.54 

12.294 

30 

881.88 

67.470 

30 

1402.0 

169.04 

40 

383.91 

12.847 

40 

890.41 

68.774 

40 

1410.9 

171.15 

50 

392.28 

13.413 

50 

898.95 

70.091 

50 

1419  7 

173.27 

8 

400.66 

13.991 

18 

907.49 

71.421 

28 

1428.6 

175.41 

10 

409.03 

14.582 

10 

916.03 

72.764 

10 

1437.4 

177.55 

20 

417.41 

15.184 

20 

924.58 

74.119 

20 

1440.3 

179.72 

30 

425.79 

15.799 

30 

933.13 

75.483 

30 

1455.1 

181.89 

40 

434.17 

16.426 

40 

941.69 

76.869 

40 

1464  0 

184  08 

50 

442.55 

17.065 

50 

950.25 

78.264 

50 

1472.9 

186  .'29 

9 

450.93 

17.717 

19 

958.81 

79.671 

29 

1481.8 

188.51 

10 

459.32 

18.381 

10 

967.38 

81.092 

10 

1490.7 

190.74 

20 

467.71 

19.058 

20 

975.96 

82.525 

20 

1499.6. 

192.99 

30 

476.10 

19.746 

30 

984.53 

83.972 

30 

1508.5 

195  25 

40 

484.49 

20.447 

40 

993.12 

85.431 

40 

1517  4 

197.53 

50 

492.88 

21.161 

50 

1001.7 

86.904 

50 

1526.3 

199  82 

10 

501.28 

21.887 

20 

1010.3 

88.389 

30 

1535  3 

202.12 

10 

509.68 

22.624 

10 

1018.9 

89.888 

10 

1544.2 

204.44 

20 

518.08 

23.375 

20 

1027.5 

91.399 

20 

1553.1 

206.77 

30 

526.48 

21.138 

30 

1036.1 

92.924 

30 

1562.1 

209  12 

40 

534.89 

24.913 

40 

1044.7 

94.462 

40 

1571  0 

211  48 

50 

543.29 

25.700 

50 

1053.3 

96.013 

50 

1580.0 

213.86 

255 


TABLE  II.— TANGENTS  AND  EXTERNALS  TO  A  1°  CURVE. 


j 

Angle. 

Tan- 
|   gent. 

BST  1** 

Tan- 
gent. 

Exter- 
nal. 

Angle. 

Tan-      Exter- 
gent.         nal. 

I. 

T. 

E. 

I. 

T. 

E. 

I. 

T. 

E. 

31° 

15H9.0 

216.25       41° 

2142.2 

387.38 

51°          2732.9 

618.39 

10 

1598.0 

818*66    !          10' 

2151.7 

390.71 

10'    2743.1 

622.81 

20 

1606.9 

221.08    1          20 

•  2161.2 

394.06 

20      2753  4 

627.24 

30 

1615.9 

223.51    1          30 

2170.8 

397.43 

30 

2763.7 

631  .69 

40 

1624.9 

225.96              40 

:  2180.3 

400.82 

40     2773.9       636  17 

50 

1633.9 

228.42 

50 

2189.9 

404.22 

50      2784  2        fU<)  (5(5 

32 

1643.0 

230.90 

42 

2199.4 

407.64       52 

2794.5        615  17 

10 

1652.0 

233.39 

10 

2209.0 

411.07     .         10 

2804.9        649.70 

20 

1661.0 

235.90 

20 

2218.6 

414.52              20      2815.2       654.25 

30 

1670.0 

238.43  i 

30 

2228.1 

417.99              30      2825  6        658  83 

40 

1679.1 

240.96 

40 

'  2237.7 

421.48  I!          40 

2835.9        663.42 

50 

1688.1 

243.52 

50 

!  2247.3 

424.98              50 

2846.3 

668.03 

33 

1697.2 

246.08  ' 

43 

2257.0 

428.50       53 

2856.7 

672.  66 

10 

1706.3 

248.66 

10 

2266.6 

432.04              10 

2867.1        677.32 

20 

1715.3 

251.26 

20 

8276.2 

435.59  i           20 

2877.5       681.99 

30 

1724.4 

253.87 

30 

2285.  !) 

439.16  i!          30     2888.0 

686.68 

40 

1733.5 

256.50 

40 

2295  6 

422.75  :|          40     2898.4 

691  .40 

50 

1742.6 

259.14    '          50 

2805!  2 

446.35              50  i  2908  9 

696.13 

34 

1751.7 

261.80      44 

2314.  9 

449.98       54         i  2919.4 

700.89 

10 

1760.8 

264.47    '          10 

2324.6 

453.62              10  !  29..".)  9 

705  6(5 

20 

1770.0 

fli7.ll>              20 

2334.3 

457.27  i 

20  i  2940.4 

710.46 

30 

1779.1 

26!).  8(5               30 

2344.1 

460.95 

30  i  2951.0 

715.28 

40 

1788.2 

272.58              40 

2353.8 

464.64 

40  i  2961.5 

720.11 

50 

1797.4 

275.31  ; 

50 

2363.5 

468.35              50     2972.1 

724  .  97' 

35 

1806.6 

27'8.05 

45 

2373.3 

472.  08       55            2982.7 

729.85 

10 

1815.7 

280.82 

10 

2383.1 

475.82              10 

2993.3 

734.76 

20 

1824.9 

283.60 

20 

2392.8 

479.59    :          20     3003.9 

739.68 

30 

1834.1 

286.39 

30 

2402.6 

483.37 

30  1  3014.5 

744.62 

40 

1843.3 

289.20 

40 

2412.4  i 

487.17  ,! 

40 

3025.2 

749.59 

50 

1852.5 

292.02              50 

2422.3 

490.98  i           50 

3035.8 

754.57 

36 

1861.7 

294.86 

46 

2432.1 

494.82       56 

3046.5 

759.58 

10 

1870.9 

297.72 

10 

2441.9 

498.67              10 

3057.2       764.61 

^20 

1880.1 

300.59 

20 

2451.8 

502.54 

20 

3067.9        769  66 

30 

1889.4 

303.47 

30 

2461.7  1 

506.42 

30 

3078.7  I     774.73 

40 

1898.6 

306.37 

40 

2471.5 

510.33              40 

3089.4       779.83 

50 

1907.9 

309.29 

50 

2481.4  ; 

514.25  : 

50 

3100.2       784.94 

37 

1917.1 

312.22       47 

2491.3 

518.20  |    57 

3110.9       790.08 

10 

1926.4 

315.17              10 

2501.2 

522.16  '|          10     3121.7  j     795.24 

20 

1935.7 

318.13    !          20 

2511.2 

526.13    '          20  i  3132.6       800.42 

30 

1945.0 

321.11              30 

2521.1 

5:30.13    !          30     3143.4  1     805.62 

40 

1954.3 

324.11              40 

2531  .  1 

534.15    i          40  j  3154.2  '>     810.85 

50 

1963.6 

327.12              50 

2541.0 

538.18               50     3165.1        816  10 

38 

1972.9 

330.15       48 

2551.0 

542.23       58         i  3176.0       821.37 

10 

1982.2 

333.19    !          10 

2561.0 

546.30    !           10  1  3186.9        826.  G6 

20 

1991.5 

886.25              20 

2571.0 

550.39 

20 

3197.8        831.98 

30 

2000.9 

339.32 

30 

2581.0 

554.50 

30     3208.8  j     837.31 

40 

2010.2 

342.41 

40 

2591.1   i 

558.63              40     3219.7       842.67' 

60 

2019.6 

345.52 

50 

2601.1   ! 

562.77    i          50 

3230.7       848.06 

39            2029.0 

348.64    i  49 

2611.2 

566.94    '.  59           3241.7  '     853.46 

10     2038.4 

351.78    !          10 

2621.2 

571.12 

10  |  3252.7       858.89 

20 

•  2047.8 

354.94    1          20 

2631.3 

575.32    |          20  !  3263.7       864.34 

30 

2057.2 

358.  .11              30 

2641.4 

579.54              30  '  3274.8  '.     869.82 

40 

2066.6 

361.29              40 

2651.5 

583.78    1          40     3285.8       875.32 

50 

2076.0 

364.50              50 

2661.6 

588.04 

50     3296.9       880.  HI 

40 

2085.4 

367.72       50 

2671.8 

592.32 

60           SJ08.0       886.38 

10 

2094.9 

370.!  IT)               10 

2681.9  , 

596.62 

10  !  3319.1       891.95 

20 

2104.3 

374.20    .          20 

2692.1   ! 

600.93              20     8330.3       897.54 

30 

2113.8 

377.47              30 

2702.3  | 

605.27 

30  i  3341.4       903.15 

40 

2123.3 

380.76              40 

2712.5 

609.62 

40     3352.6       908.79 

50 

2132.7 

384.06 

50 

2722.7 

614.00  i           50  i  3363.8       914.45 

256 


TABLE  II.— TANGENTS  AND  EXTERNALS  TO  A  1°  CURVE. 


Angle. 

Tan- 
gent. 

Exter- 
nal. 

Angle. 

Tan- 
gent. 

Exter- 
nal. 

Angle. 

Tan- 
gent. 

Exter- 
nal. 

I. 

T. 

E. 

I. 

T. 

E. 

T. 

T. 

E. 

61° 

3375.0 

920.14 

71° 

4086.9 

1308.2 

81° 

4893.6 

1805.3 

10' 

3386.3 

925.85 

10' 

4099.5 

1315.6 

10' 

4908.0 

1814.7 

20     3397.5 

931.58 

20  !  4112.1      1322.9 

20 

4922.5 

1824.1 

30  !  3408.8  |     937.34 

30     4124.8      1330.3 

30     4937.0 

1833.6 

40     3420.1  !     943.12 

40     4137.4      1337.7 

40      4951.5 

1843.1 

30 

3431.4  i     948.92 

50 

4150.1      1345.1 

50  i  4966.1 

1852.6 

62 

3442.7 

954.75 

72 

4162.8      1352.6 

82         ;  4980.7 

1862.2 

10     3454.1 

960.60 

10 

4175.6  ''  1360.1 

10  i  4995.4 

1871.8 

20     3405.4 

966.48 

20 

4188.5  !  1367.6 

20      5010.0 

1881.5 

30     3476.8 

972.38  : 

30     4201.2      1375.2 

30      5024.8 

1891.2 

40     3488.3 

978.31 

40 

4214.0      1382.8 

40      5039.5 

1900.9 

50 

3499.7 

984.27  :           50 

4226.8  :   1390.4 

50      5054.3 

1910.7 

63 

3511.1 

990.24       73 

4239.7      1398.0 

83            5069.2 

1920.5 

10  |  3522.  (5 

996.24              10 

4252.6      1405.7 

10     5084.0 

1930.4 

20  !  3534.1 

1002.3 

20 

4265.6      1413.5 

20 

5099.0      1940.3 

30     a545.6 

1008.3 

30 

4278.5      1421.2 

30     5113.9     1950.3 

40     3557.2 

1014.4                40 

4291.5  !  1429.0 

40  i  5128.9      1960.2 

50     3568.7 

1020.5      '          50 

4304.6      1436.8 

50      5143.9      1970.3 

64           3580.3 

1026.6      i  74 

4317.6     1444.6 

84         i  5159.0     1980.4 

10     3691.9 

1032.8                10 

4330.7 

1452.5              10  :  5174.1      1990.5 

20     3603.5 

1039.0                20 

4343.8 

1460.4              20  !  5189.3  I  2000.6 

30     3615.1 

1045.2 

30     4356.9 

1468.4 

30  '  5204.4      2010.8 

40  I  3626.8 

1051.4 

4jO 

4370.1 

1476.4 

40      5219.7     2021.1 

50     3638.5 

1057.7 

50 

4383.3 

1484.4 

50      5234.9     2031.4 

65 

3650.2 

1063.9 

75 

4396.5 

1492.4 

85 

5250.3     2041.7 

10 

3661.9 

1070.2 

10 

4409.8      1300.5 

10      5265.6      2052.1 

20  !  3673.7     1076.6 

20     4423.1      1508.6 

20      5281.0     2062.5 

30 

3685.4 

1082.9 

30  1  4436.4 

1516.7 

30  i  5296.4  i  2073.0 

40 

3697.2 

1089.3 

40  i  4449.7 

1524.9 

40      5311.9     2083.5 

50 

3709.0 

1095.7 

50 

4463.1 

1533.1 

50      5327.4      2094.1 

66           3720.9 

1102.2 

76 

4476.5      1541.4 

86            5343.0  !  2104.7 

10     3732.7 

1108.6     i 

10     4489.9     1549.7          ~   10     5358.6  ;  2115.3 

20 

3744.6 

1115.1 

20     4503.4      1558.0              20      5374.2      2126.0 

30 

3756.5 

1121.7    ! 

30  1  4516.9     1566.3             30     5389.9 

2136.7 

40 

3768.5 

1128.2     ! 

40     4530.4     1574.7 

40      5405.6 

2147.5 

50 

3780.4 

1134.8 

50 

4544.0  ;  1583.1 

-       50 

5421.4 

2158.4 

67 

3792.4 

1141.4 

77 

4557.6  !  1591.6 

87 

5437.2 

2169.2 

10 

3804.4 

1148.0     ! 

10 

4571.2  !  1600.1 

10 

5453.1 

2180.2 

20 

3816.4 

1154.7 

20     4584.8      1608.6 

20 

5469.0 

2191.1 

30 

3828.4 

1161.3 

30  1  4598.5 

1617.1 

30 

5484.9 

2202.2 

40 

3840.5 

1168.1 

40 

4612.2 

1625.7 

40 

5500.9 

2213.2 

50 

3852.6 

1174.8 

50 

4626.0 

1634.4 

50 

5517.0 

2224.3 

68 

3864.7 

1181.6 

78 

4639.8 

1643.0 

88 

5533.1 

2235.5 

10 

3873.8     1188.4 

10 

4653.6 

1651.7 

10 

5549.2 

2246.7 

20  i  3889.0     1195.2 

20 

4667.4 

1660.5 

20 

5565.4 

2258.0 

30 

3901.2      1202.0 

30 

4681.3 

1669.2- 

30 

5581.6 

2269.3 

40 

3913.4  |  1208.9 

40 

4695.2 

1678.1 

40 

5597.8 

2280.6 

50 

3925.6 

1215.8 

50 

4709.2 

1686.9 

50 

5614.2 

2292.0 

69 

3937.9 

1222.7 

79 

4723.2 

1695.8 

89 

5630.5 

2308.5 

10 

3950.2 

1229.7 

10 

4737.2 

1704.7 

10 

5646.9 

2315.0 

20 

3962.5 

1236.7 

20 

4751.2 

1713.7 

20 

5663.4 

2326.6 

30     3974.8 

1243.7 

30 

4765.3 

1722.7 

30 

5679.9 

2338.2 

40     3987.2 

1250  8 

40 

4779.4 

1731.7 

40 

5696  4 

2349.8 

50     3999.5      1257.9                50 

4793.6 

1740.8 

50 

5713.0 

2361.5 

70 

4011.9      1265.0         80 

4807.7 

1749.9 

90 

5729.7 

2373.3 

10  ;  4024.4  |  1272.1     !            10 

4822.0 

1759.0 

10 

5746.3     2385.1 

20  '  4036.8  i  1279.3                20      4836.2      1768  2 

20 

5763.1   :  2397.0 

30  i  4049.3      1286.5      i          80      4850.5  I  1777.4  i            30  j  5779  9  '  2408.9 

40     4061.8 

1293.6 

40  i  4864.8 

1786.7 

40     5796.7 

2420.9 

50  1  4074.4 

1300.9 

50      4879.2 

1796.0 

50     5813.6 

2432.9 

257 


TABLE  II.— TANGENTS  AND  EXTERNALS  TO  A  1°    CURVE. 


Angle. 

Tan- 
gent. 

Ex- 
ternal. 

Angle. 

Tan- 
gent. 

teSfal.  i|  Angle. 

Tan- 
gent. 

Ex- 
ternal 

I. 

T. 

E. 

I. 

T. 

E. 

*' 

T. 

E. 

91° 

5830.5 

2444.9 

97 

6476.2  '  2917.3 

103 

7203 

.2 

3474.4 

10' 

5847.5 

245" 

r.i 

1 

0 

64 

95.2      2!)31 

(5 

10 

7224 

_7 

34 

1)1.3 

20 

5864.6 

2469.3 

20     6514.3      2945.9 

20 

7246.3 

3508.2 

ao 

5881.7  i  2481 

.5 

c 

10    o.: 

33.4      2960 

3 

30 

7268 

.0 

33 

25.2 

40 

5898.8  !  249: 

5.8 

< 

K)     65 

52.6      2974 

40 

728? 

.8 

8! 

42.4 

50 

5916.0  i  250t 

5.1 

£ 

ti 

Gi 

71.9      2989 

2 

50 

7811 

_  / 

3T 

59.6 

92 

5933.2      2518.5 

98 

6591.2      3(X)3 

8 

104 

7838 

.6 

3576.8 

10 

5950.5      2531.0 

10     6610.6     3018.4 

10 

7855 

.15 

3594.2 

20 

5967.9  i  254: 

5.5 

s 

!0  !  01 

.80.1      3033 

1 

20 

7377 

.8 

31 

11.7 

80 

5985.3      255( 

5.0 

I 

50 

61 

49.6  i  3047 

i) 

30 

7399 

9 

:',( 

29.2 

40 

6002.7      256* 

4 

0 

6b 

69.2  i  3062 

8 

40 

7422 

.2 

.'if 

>46.8 

50 

6020.2      2581.3 

50 

6688.8      3077 

7 

50 

7444 

.6     3664.5 

93 

6037.8     2594.0 

99 

6708.6 

3092.7 

105 

746! 

.0 

a 

>82.3 

10 

6055.4      260( 

5.8 

1 

0 

67 

28.4      3107 

7  i 

10 

7489 

.0 

V 

100.2 

20 

6073.1      261< 

).7 

$ 

0 

67 

48.2  !  3122 

9 

20 

7512 

.2 

3' 

"18.2 

30 

6090.8     2632.6 

30 

6768.1  !  3138 

1 

30 

7534.9 

3736.2 

40 

6108.6      264; 

5.5 

- 

0 

07 

88.1  i  3153 

3 

40 

7557 

.7 

Of 

'54.4 

50 

6126.4      265* 

5.5 

0 

6£ 

,08.2  !  3168 

50 

7580 

.5 

3' 

72.0 

94 

6144.3      267 

1.6 

100 

6828.3      3184 

1 

106 

7603.5 

3791.0 

10 

6162.2 

2(58- 

L7 

] 

0 

68 

48.5  i  3199 

6 

10 

7620 

.0 

3* 

909.4 

20 

6180.2 

269 

".9 

£ 

0 

0> 

168.8  l  3215 

1   ; 

20 

7649 

.7 

3! 

•127.9 

30 

6198.3  I  2711.2 

30 

68 

Sit.  2      32:30.8 

30 

7672.9 

3846.5 

40 

6216.4  j  2724.5 

40 

6909.6      3240.5 

40 

70%.  3 

3865.2 

50 

6234.6  |  2737.9 

50 

6930.1 

3262.3 

50 

7719.7 

3884.0 

95 

6252.8     2751.3 

101° 

6£ 

50.6 

3278.1 

107 

7743.2 

3902.9 

10 

6271.1 

276- 

1.8 

] 

0' 

6t 

71.3 

3294 

1 

10 

7700 

.8 

3 

J21.9 

20 

6289.4 

2778.3 

J 

50 

6992.0 

3310.1 

20 

7790.5 

3940.9 

30 

6307.9 

279S 

J.O 

( 

n 

7C 

12.7      3326 

1 

30 

7814 

.3 

3 

>60.1 

40 

6326.3      2805.6 

40 

7033.6      3342 

3 

40 

7838.1 

3979.4 

50 

6:344.8      281( 

).4 

j 

>o 

7C 

54.5      3358 

5 

50 

7'80^ 

.1 

3 

398.7 

96 

6363.4     283, 

J.2 

102  ^ 

7075.5      3374 

9 

108 

7886.2 

4018.2 

10 

6382.1  !  284 

r.o 

1 

0 

7( 

96.6      3391 

2 

10 

791C 

.4 

4) 

J37.8 

20 

6400.8     286 

l.O 

» 

71 

17.8  i  3407 

7 

20 

79:34 

.(i 

4 

357.4 

30 

6419.5 

2875.0 

80 

7139.0  !  3424 

8 

30 

7839.0 

4077.2 

40 

6438.4 

288 

).0 

i 

0 

7] 

60.3      3440 

9 

40 

7983 

.5 

4 

D97.1 

50 

6457.3 

2903.1 

50 

7181.7     3457.6 

50 

8008,0 

4117.0 

CORRECTIONS  FOR  TANGENTS  AND  EXTERNALS. 

FOR  TANGENTS,  ADD 

FOR  EXTERNALS, 

ADD 

Ang 

5°      10° 

15° 

20° 

25° 

30° 

Ang 

5° 

10°     15° 

20° 

25° 

30° 

I- 

Cur.  Cur.  Cur.  1  Cui 

•.  Cur. 

Cur. 

I  !• 

Cur. 

Cur.  Cur. 

Cur. 

Cur. 

Cur. 

10° 

.03      .06 

.09 

.] 

J     .16  i    .19 

10° 

001 

.003     .004  ~T006~ 

.007 

.008 

20 

.06      .13 

.19 

5      .32 

.39 

20 

.006 

.011      .017 

.022 

.O1 

JS 

.034 

30 

.10      .19 

.29 

'.3 

)      .49 

.59 

30 

.013 

.025     .0381   .051 

;.-> 

.078 

40 

.13      .26 

.40 

.5, 

i      .67 

80 

40 

.023 

.046     .0701   .093 

'.117 

.141 

50 

.17      .34 

.51 

.6i 

J      .85 

1  02 

!  50 

.037 

.075      .116     .151 

.1 

s<) 

.227 

60 

.21      .42 

.63 

.84    1.05 

1.27 

60 

.056 

.112      .168     .225 

.2 

S3 

.340 

70 

.25      .51 

.76 

1.02    1.28 

1.54 

70 

.oso 

.159     .240     .321 

.4 

).", 

.485 

80 

.30      .61 

.91 

1.2211.53 

1.84 

80 

.110 

.220      .332!   .445 

.558 

.071 

90 

.36      .72 

1.09 

1.45    1.83 

2.20 

90 

.149 

.299      .450     .603 

.7 

56 

.910 

100 

.43      .86 

1.30 

1.74   2  is 

2.62 

100 

.200 

.401      .604     .809 

1.0 

15 

1.221 

110 

.51    1.03 

1.56 

2  08   2.61 

3.14 

110 

.268 

.536      .806  1.082 

1.3 

55 

1.683 

120 

.62    1.25 

1.93 

2.5 

2    3  16 

3.81 

120 

,300 

.72111.086  1.456 

1.825 

2.197 

258 


TABLE  III.-TANGENTIAL  OFFSETS  100  FT.  ALONG  THE  CURVE. 


Deg.  of 

0' 

10' 

20' 

30' 

40' 

50' 

Curve. 

0° 

0.000 

0.145 

0.291 

0  * 

136 

0.582 

0.727 

1° 

0.873 

1.018 

1.164 

lii 

$09 

1.454 

- 

.600 

2° 

1.745 

1.89 

1 

2.036 

2. 

81 

2.327 

S 

.472 

3° 

2.618 

2.763 

2.908 

3.054 

3.199 

3.345 

4° 

3.490 

3.03 

3.781 

3. 

)26 

4.071 

4 

.217 

5° 

4.362 

4.507 

4.653 

4  f 

-98 

4.943 

5.088 

6° 

5.234 

5.37 

) 

5.524 

5. 

369 

5.814 

£ 

.960 

6.105 

6.250 

6.395 

6.540 

6.685 

( 

.831 

8° 

6.976 

7.12 

I 

7.266 

7. 

111 

7.556 

r 

.701 

9° 

7.846 

7.99 

I 

8.136 

8  ' 

>81 

8.426 

e 

t.57i 

10° 

8.716 

8.86 

} 

9.005 

9J50 

9.295 

c 

.440 

11° 

9.585 

9.729 

9.874 

10.019 

10.164 

10.308 

12° 

10.453 

10.59 

~ 

10.742 

10. 

387 

11.031 

1] 

.176 

13° 

11.320 

11.465 

11.609 

11.754 

11.89S 

12.043 

14° 

12.187 

12.33 

1 

12.476 

12. 

520 

12.764 

IS 

.908 

15° 

13.053 

13.19 

13.341 

13. 

185 

13.62E 

U 

1.773 

16° 

13.917 

14.061 

14.205 

14.349 

14.49S 

14.637 

17° 

14.781 

14.92 

5 

15.069 

15. 

>12 

15.356 

If 

..500 

18° 

15.643 

15.787 

15.931 

16.074 

16.218 

16.361 

19° 

16.505 

16.648 

16.792 

16. 

?35 

17.078 

17.222 

20° 

17.365 

17.50 

3 

17.651 

17. 

17.937 

1£ 

5.081 

21° 

18.224 

18.36 

r 

18.509 

18. 

552 

18.795 

11 

i.938 

22° 

19.081 

19.224 

19.366 

19.509 

19.652 

» 

>.794 

23° 

19.937 

20.07 

i 

20.222 

20. 

364 

20.507 

2( 

).649 

24° 

20.791 

20.933 

21.076' 

21.218 

21.36C 

21.502 

TABLE  IV.—  MID-ORDINATES  TO  A  100-FT.  CHORD. 

Deg. 

of 

0           1 

2 

3          4 

5 

6           7  ' 

8 

9 

Curve. 

0° 

0.000    0.21* 

5    0.436 

0.655    0.873 

1.091 

1.309    1.528 

1.746 

1.965 

10° 

2.183    2.405 

>    2.620 

2.839    3.058 

3.277 

3.496    3.716 

3.935 

4.155 

20° 

4.374    4.59' 

I    4.814 

5.035    5.255 

5.476 

5.697    5.918 

6.139 

6.360 

259 


TABLE  V.-LONG  CHORDS. 


Degree 

Actual 
Arc, 

LONG  CHORDS. 

of 
Curve. 

One 
Station. 

Stations. 

3 

Stations. 

4          |         5 

Stations.  |  Stations. 

6 

Stations. 

0°10' 

100.000 

200.000 

299.999 

399,998 

499.996 

599.993 

20 

.000 

199.999     1       299.997 

399.992 

499.983 

599.970 

30 

.000 

199.998     :       299.992 

399.981         499.962 

599.9:33 

40 

.001 

199.997           299.986 

399.966     ;     499.932 

599.882     i 

50 

.001 

199.  095 

299.979 

399.947     i     499.894 

599.815 

1 

100.001 

199.992 

299.970         399.924         499.848 

599.733 

10 

.002 

199.990     !       299.959         399.896 

499.793     j     599.637 

20 

.002 

199.986           299.946     1     399  .'865 

499.729     i     599.526 

30 

.003 

199.983 

299.932 

399.829 

499.657         599.401 

40 

.003 

199.979 

299.915 

399.789 

499.577          599.260 

50 

.004 

199.974 

299.898 

399.744 

499.488         599.105 

2 

100.005 

199.970 

299.878 

399.695 

499.391 

598.934 

10 

.006 

199.964 

299.857 

399  .  643 

499.285 

59H.750 

20 

.007 

199.959 

299.834 

399.586 

499.171     .     598.550 

30 

.008 

199.952 

299.810 

399.524 

499.049         598.336 

40 

.009 

199.946 

299.783 

399.459 

498.918         598.106 

50 

.010 

199.939 

299.756 

399.389         498.778 

597.862 

3 

100.011 

199.931           299.726 

399.315     !     498.630 

597.604 

10 

.013 

199.924           299.695 

399.237 

498.474     I     597.331 

20 

.014 

199.915           299.662 

399.154 

498.309         597.043 

30 

.015 

199.907     1"     299.627 

399.068 

498.136         596.740 

40 

•017 

199.898 

299.591 

398.977 

497.955     ;     596.423 

50 

.019 

199.888 

299.553 

398.882 

497.765 

596.091 

4 

100.020 

199.878 

M0.528 

398.782 

497.566 

595.744 

10 

.022 

199.868 

299.471 

398.679 

497.360 

595.383 

20 

.024 

199.857 

299.428 

398.571 

497.145 

595.007 

30 

•  .026 

199.846 

299.383 

398.459 

496.921         594.617 

40 

.028 

199.834 

299.337 

398.343 

496.689 

594.212 

50 

.030 

199.822 

299.289 

398.223 

496.449 

593.792 

5 

100.032 

199.810 

299.239 

398.099 

496.201 

593.358 

10 

.034 

199.797 

299.187 

397.970 

495.944 

592.909 

20 

.036 

199.783 

299.134 

397.837 

495.678 

592.446 

30 

.038 

199.770 

299.079 

397.700 

495.405 

591.968 

40 

.041 

199.756 

299.023 

397.559 

495.123 

591.476 

50 

.043 

199.741 

298.964 

397.413 

494.832 

590.970 

6 

100.046 

199.726 

298.904 

397.264 

494.534 

590.449 

10 

.048 

199.710 

298.843 

397.110 

494.227 

589.913 

20 

.051 

199.695 

298.779 

396.952 

493.912 

589.364 

30 

.054 

199.678 

298.714 

396.790 

493.588 

588.800 

40 

.056 

199.662 

298.648 

396.623 

493.257 

588.221 

50 

.059 

199.644 

298.579 

396.453 

492.917 

587.628 

7 

100.062 

199.627 

298.509 

396.273 

492.568 

587.021 

10 

.065 

199.609 

298.438 

396.099 

492.212 

586.400 

20 

.068 

199.591 

298.364 

395.916 

491.847 

685.766 

30 

.071 

199.572 

298.289 

395.729 

491.474 

585.115 

40 

.075 

199.553 

298.212 

395.538 

491.093 

584.451 

50 

.078 

199.533 

298.134 

395.342 

490.704 

583.773 

8 

100.081 

199.513 

298.054 

395.142 

490.306 

sas.ofii 

10 

.085 

199.492 

297.972 

394.938 

489.900 

582.375 

20 

.088 

199.471 

297.888 

394.731 

489.486 

581.654 

30 

.092 

199.450 

297.803 

394.518 

489.064 

580.920 

40 

.095 

199.428 

297.716 

394.302 

488.634 

580.172 

50 

.099 

199.406 

297.628 

394.082 

488.196 

579.409 

9 

100.103 

199.383 

297.538 

393.857 

487.749 

578.633 

10 

.107 

199.360 

297.446 

393.629 

487.294 

577.843 

20 

.111 

199.887 

297.352         393.396 

486.832 

577.039 

30 

.115 

199.313 

297.257         393.159 

486.361 

576.222 

40 

.119 

199.289 

297.160 

392.918 

485.882 

575.390 

50 

.123 

199.264 

297.062 

392.673 

485.395 

574.545 

10 

100.127 

199.239 

29G.962 

392.424 

484.900 

573.686 

TABLE  V.— LONG  CHORDS. 


Degree 
of 
Curve. 

Actual 
Arc, 

One 
Station. 

LONG  CHORDS. 

2 

Stations. 

3 

Stations. 

4 

Stations. 

5 

Stations. 

6 

Stations. 

10°10'i     100.131           199.213            296.860 

392.171 

484.397 

572.813 

20             .136  !         199.187             296.756 

391.914 

483.886 

571.926 

30 

.140  i        199.161             296.651 

391.652 

483.367 

571.027 

40 

.145           199.1:34             296.544 

391.387 

482.840 

570.113 

50  j           .149           199.107     j        296.436 

391.117 

482.305 

569.186 

11          !     100.154 

199.079     i        296.325 

390.843 

481.762 

568.245 

10             .158  i         199.051 

296.214 

390.565 

481.211 

567.292 

20 

.163  j        199.023 

296.100 

390.284 

480.653 

566.324 

30 

.168  '         198.994             295.985 

389.998 

480.086 

565.343 

40 

.173 

198.964 

295.868 

389.708 

479.511 

564.349 

50             .178 

198.935 

295.750 

389.414 

478.929 

563.341 

12              100.183 

198.904 

295.629 

389.116 

478.338 

562.321 

10  !          .188           198.874 

295.508 

388.814 

477.740 

561.287 

20  i          .193           198.843 

295.384 

388.508 

477.135 

560.240 

30  1          .199           198.811 

295.259 

388.197 

476.521 

559.180 

40  !          .204  !        198.779 

295.132 

387.883 

475.899 

558.107 

50  !          .209           198.747 

295.004 

387.565 

475.270 

557.020 

13              100.215           198.714 

294.874 

387.243 

474.633 

555.921 

10             .220          198.681 

294.742 

386.916 

473.988 

554.809 

20             .226           198.648 

294.609 

386.586 

473.336 

553.684 

30             .232 

198.614 

294.474 

386.252 

472.675 

552.546 

40  !           .237 

198.579 

294.337 

385.914 

472.007 

551.395 

50 

.243 

198.544 

294.199 

385.572 

471.332 

550.232 

14 

100.249  i        198.509 

294.059 

385.225 

470.649 

549.056 

10 

.255           198.474 

293.918 

384.875 

469.958 

547.867 

20 

.261           198.437 

293.774 

384.521 

469.260 

546.666 

30 

•  267  !        198.401 

293.629 

384.163 

468.554 

545.452 

40 

•  274 

198.364 

293.483 

383.801 

467.840 

544.226 

50 

•  280 

198.327 

293.335 

383.435 

467.119 

542.987 

15 

100.286  1        198.289 

293.185 

383.065 

466.390 

541.736 

10 

.292           198.251 

293.034 

382.691 

465.654 

540.472 

20 

.299 

198.212 

292.881 

382.313 

464.911     !     539.196 

30 

.306 

198.173 

293.786 

381.931 

464.160     j     537.908 

40  1          .312 

198.134 

292.570 

381.546 

463.401 

536.608 

50             .319 

198.094 

292.412 

381.156 

462.635 

535.296 

16;            100.326  i        198.054 

292.252 

380.763' 

461.862 

533.972 

10 

.333 

198.013 

292.091 

380.365 

461.081 

532.635 

20 

.339 

197.972 

291.928 

379.964 

460.293 

531.287 

30 

.346 

197.930 

291.764 

379.559 

459.498 

529.927 

40 

.353 

197.888 

291.598 

379.150 

458.695 

528.555 

50 

.361 

197.846 

291.430 

378.737 

457.886 

527.171 

17 

100.368 

197.803 

291.261 

378.320 

457.069 

525.776 

10 

.375 

197.760 

291.090 

377.900 

456.244 

524.369 

20 

.382 

197.716 

290.918 

377.475 

455.413 

522.950 

30 

.390 

197.672 

290.743 

377.047 

454.574 

521.519 

40 

.397 

197.628 

290.568 

376.615 

453.728 

520.078 

50 

.405 

197.583 

290.390 

37'6.179 

452.875 

518.625 

18 

100.412 

197.538 

290.211 

375.739 

452.015 

517.160 

10 

.420 

197.492 

290.031 

375.295 

451.147 

515.685 

20 

.428 

197.446 

289.849 

374.848 

450.373 

514.198 

30 

.436 

197.399 

289.605 

374.397 

449.392 

512.699 

40 

.444 

197.352 

289.479 

373.942         448.504 

511.190 

,«   50 

.452 

197.305 

289.292 

873.483         447.608 

509.670 

19 

100.460 

197.256 

289.104 

373.021         446.7'06 

508.139 

10 

.468 

197.209 

288.913 

372.554 

445.797         506.597 

20 

.476 

197.160 

288.722 

372.084 

444.881         505.043 

30 

.484           lOr.lll 

288.528 

371.610         443.957         503.479 

40             .493 

197.082            288.333 

371.133     I     443.028         501.905 

...    50             .501 

197.012    i        288.137 

370.652         442.091 

500.320 

20             100.510           196.962            287.939 

370.167         441.147 

498.724 

261 


TABLE  VI.-MID-ORDINATES  TO   LONG  CHORDS. 


Degree 
of 
Curve. 

1 

Station. 

g 

Stations. 

3 

Stations. 

4 

Stations. 

5 

Stations. 

6 

Stations. 

o°  i<y 

.036 

.145 

.327 

.582 

.909 

1  309 

20            .073 

.291 

.654 

1.164 

1.818 

2.618 

30    i        .109 

.436              .982 

1.745 

2.727 

3.926 

40             .145 

.582 

1.309 

2.327 

3.636 

5.235 

50 

.182 

.727 

1.636 

2.909 

4.545 

6.544 

1 

.218 

.873 

1.903 

3.490 

5.453 

7  852 

10 

.255 

1.018            2.291 

4.072 

6.362 

9.160 

20 

.291 

1.164             2.618              4.654 

7.270 

10.468 

30 

.327 

1.309 

2.945 

5.235 

8.179 

11.775 

40 

.364 

1.454 

3.272 

5.816 

9.087 

1     13.082 

50 

.400 

1.600 

3.599 

6.398 

9.994 

14.389 

2 

.436 

1.745 

3.926 

6.979 

10.902 

15.694 

10 

.473 

1.891 

4.253 

7.560 

11.809 

17.000 

20 

.509            2.036 

4.580 

8.141 

12.716 

i     18.304 

30 

.545            2.181 

4.907 

8.722 

13.623 

19.608 

40 

.582            2.327 

5.234 

9.303 

14.529 

20.912 

50 

.618            2.47'2 

5.561 

9.883 

15.435 

22.214 

3 

.654 

2.618 

5.888 

10.464 

16.341 

!     23.516 

10 

.691 

2.763 

6.215 

11.044 

17.246 

'     24.817 

20             .727 

2.908 

6.542 

11.624             18.151 

26.117 

30    !         .703 

3.054            6.868 

12.204             19.055 

27.416 

40             .800 

3.199 

7.195 

12.784             19.959 

i     28.714 

50    J         .836 

3.345 

7.522 

13.363 

20.863 

30.012 

4                    .872 

3.490      I      7.848 

13.943 

21.766 

31.308 

10 

.909 

3.635 

8.175 

14.522 

22.668 

32.603 

20     !         .945             3.781 

8.501 

15.101 

23.570 

&3.  896 

30    j         .982 

3.926            8.828 

15.680 

24.471 

35.189 

40    !      1.018 

4.071 

9.154 

16.258 

25.372 

36.480 

50 

1.054 

4.217             9.480 

16.837 

26.272 

37.770 

5 

1.091 

4.362             9.807 

17.415 

27.171 

39.059 

10 

1.127 

4.507           10.133 

17.992 

28.070 

40.346 

20 

1.164 

4.653       1     10.459 

18.570 

28.968 

41.631 

30 

1.200 

4.798           10.785 

19.147 

29.866 

42.916 

40 

1.237 

4.943       1     11.111 

19.724 

30.762 

44.198 

50 

1.273 

5.088           11.436 

20.301 

31.658 

45.479 

6 

1.309 

5.234 

11.762 

20.877 

32.553 

46.759 

10 

1.346 

5.379           12.088 

21.453 

33.448 

48.037 

20 

1.382 

5.524 

12.413 

22.029 

34.341 

49.313 

30           1.418 

5.669 

12.739 

22.604 

35.234 

50.587 

40           1.455 

5.814           13.064 

23.179 

36.126 

51.860 

50 

1.491 

5.960 

13.389 

23.754 

37.017 

53.130 

7 

1.528 

6.105           13.715 

24.328 

37.907 

54.399 

10 

1.564 

6.250 

14.040 

24.902 

38.796 

55.666 

20 

1.600 

6.395 

14.365 

25.476 

39.684 

56.931 

30 

1.637 

6.540 

14.689 

26.049 

40.571 

58.193 

40 

1.673 

6.685 

15.014 

26.622 

41.458 

59.454 

50 

1.710 

6.831 

15.339 

27.195 

42.343 

60.712 

8 

1.746 

6.976 

15.663 

27.767 

43.227 

61.969 

10 

1.762 

7.121 

15.988 

28.338 

44.110 

63,223 

20 

1.819 

7.266 

16.312 

28.910 

44.992 

64.47'5 

30 

1.855 

7.411 

16.636 

29.481 

45.873 

65.784 

40 

1.892 

7.556           16.960 

30.051 

46.753     j 

66.972 

50 

1.928 

7.701           17.284 

30.621 

47.632 

68.216 

9 

1.965 

7.846 

17.608 

31.190 

48.510 

69.459 

10 

2.001 

7  991            17  932            31  7'59 

49.386 

70.699 

20 

2.037 

8'.  136 

18.255            32.328 

50.261     j 

71  .  930 

30 

2.074 

8.281 

18.578            32.8% 

51.135 

73.171 

40 

2.110 

8.426           18.902            &S.464 

52.008 

74  403 

50 

2.147 

8.571            19.225            34.031 

52.880 

75.632 

10 

2.183 

8.716 

19.548            34.597            53.750 

76.8>9 

262 


TABLE  VI.-MID-ORDINATES  TO  LONG  CHORDS. 


Degree 
of 
Curve. 

1 

Station. 

2 

Stations. 

3 

Stations. 

4 

Stations. 

6 

Stations. 

6 

Stations. 

10°  10' 

2.219            8.860 

19.870 

35.164 

54.619 

78.083 

20 

2.256 

9.005 

20.193 

35.729 

55.486 

79.305 

30 

2.293 

9.150 

20.516 

36.294     i     56.353 

80.523 

40 

2.329 

9.295 

20.838 

36.859         57.218 

81.739 

50 

2.365            9.440 

21  .  160 

37.423         58.081 

82.951 

11 

2.402     i        9.585 

21.483 

37.986 

58.943 

84.161 

10 

2.438 

9.729 

21.804 

38.549 

59.804 

85.368 

20 

2.475 

9.874 

22.126 

39.111 

60.663 

86.571 

30 

2.511 

10.019 

22.448 

39.673 

61.521 

87.772 

40 

2.547 

10.164 

22.769 

40.234         62.377 

88.969 

50 

2.584 

10.308 

23.090 

40.795         63.232 

90.164 

12 

2.620 

10.453 

23.412 

41.355         64.085 

91.355 

10 

2.657 

10.597 

23.7-33 

41.914         64.937 

92.542 

20 

2.693 

10.742 

24.053 

42.473     '     65.787 

93.727 

30 

2.730 

10.887 

24.37'4 

43.031     !     66.636 

94.908 

40 

2.766 

11.031 

24.694 

43.588     !     67.482 

96.086 

50 

2.803 

11  .  176 

25.014 

44.145     !     68.328 

97.260 

13 

2.839 

11.320 

25.334 

44.701     !     69.171 

98.431 

10 

2.876 

11.465 

25.654 

45.256         70.013             99.598 

20 

2.912 

11.609 

25.974 

45.811         70.854           100.762 

30 

2.949 

11.754           26.293 

46.365         71.692           101.922 

40 

2.985 

11.898 

26.612 

46.919         72.529 

103.079 

50 

3.022 

12.043 

26.931 

47.472         73.364 

104.232 

14 

3.058 

12.187 

27.250 

48.024 

74.197 

105.381 

10 

3.095     !      12.331           27.569 

48.575 

75.029 

106.527 

20 

3.131           12.476          27.887 

49.126         75.859 

107.669 

30 

3.168    i      12.620 

28.206 

49.676         76.687 

108.807 

40 

3.204     !      12.764 

28.524 

50.225         77.513 

109.941 

50 

3.241           12.908 

28.841           50.77.3         78.337 

111.071 

15 

3.277          13.053 

29.159           51.321         79.159 

112.197 

10 

3.314          13.197 

29.476 

51.868         79.979 

113.319 

20 

3.350          13.341 

29.794 

52.414         80.798 

114.438 

30 

3.387          13.485 

30.111 

52.959         81.614 

115.552 

40 

3.423 

13.629 

30.427 

53.504     :     82.429 

116.662 

50 

3.460 

13.773 

30.744 

54.048     |     83.241 

117.768 

16 

3.496 

13.917 

31.060 

54.591 

84.052 

118.870 

10 

3.533 

14.061 

31.376 

55.133 

84.861 

119.967 

20 

3.569 

14.205 

31.692 

55.675     i     85.667 

121.061 

30 

3.606 

14.319 

32.008 

56.215         86.471 

122.150 

.40 

3.643 

14.493 

32.323 

56.755         87.274 

123.235 

50 

3.679 

14.637 

32.638 

57.294         88.074 

124.315 

17 

3.  716 

14.781 

32.953 

57.832     i     88.872 

125.391 

10 

3.752 

14.925 

as.  267 

58.369         89.G68 

126.463 

20 

3.789 

15.069 

33.582 

58.906         90.462 

127.530 

30 

3.825 

15.212 

33.896 

59.441          91.254 

128.E23 

40 

3.862 

15.356 

34.210 

59.1)£6          92.043 

129.651 

50 

0.899 

15.500 

34.523 

60.510 

92.830 

130.704 

18 

3.935          15.643 

34.837 

61.042 

93.616 

131.753 

10 

3.972          15.787 

35.150 

61.574 

94.398 

132.797 

20 

4.008    !      15.931 

35.463 

62.106 

95.179 

133.837 

30 

4.045 

16.074 

35.775 

62.636 

95.957 

134.872 

40 

4.081 

16.218 

36.088 

63.165 

96.733 

135.902 

50 

4.118 

16.361 

36.400 

63.693 

97.506 

136.928 

19 

4.155 

16.505 

36.712 

64.221 

98.278 

137.948 

10 

4.191 

16.648 

37.023 

64.747 

99.047 

138.964 

20 

4.228 

16.792 

37.334 

65.273 

99.813 

139.975 

30 

4.265 

16  9.35 

37.645 

65.797 

100.577 

140.981 

40 

4.301 

17.078 

37.956 

66.321 

101.339 

141.982 

50 

4l  886 

17.222           38.266 

66.843 

102.098 

142.978 

20 

4.374 

17.365          38.576 

67.365 

102.855 

14,3.969 

263 


TABLE  VII. -MINUTES  IN  DECIMALS  OF  A  DEGREE. 


/ 

0' 

10" 

15' 

20" 

30" 

40" 

45" 

50" 

/ 

0 

.00000 

00278 

.00417 

.00556 

.00833 

.01111 

.01860 

.01389 

0 

1 

.01(567 

.01944 

.03088 

.02222 

.02500 

.02778 

.02917 

[08055 

1 

2 

.03333 

.03611 

.03750 

.03889 

.04167 

.04444 

.04583 

.04722   2 

3 

.05000 

.05278 

.05417 

05556 

.05833 

.U0111 

.00250 

00389   3 

4 

.06(567 

.06944 

.07083 

.07222 

.07500 

.07778 

.07917 

.08050   4 

5 

.08333 

.08011 

.08750 

.08889 

.09107 

.09444 

.09588 

.09722 

5 

6 

.10000 

.  10278 

.10417 

.10556 

.10833 

.11111 

.11860 

.11889 

(5 

7 

.11667   .11944 

.12083 

.12222 

.12500 

.12778 

.12917 

.13056  i  7 

8 

.13333 

.13611 

.18750 

.13889 

.14167 

.14444 

.14583 

.14122   8 

9 

.15000 

.15278 

.15417 

.15550 

.15833 

.10111 

.10250 

.10389   !) 

10 

.16667 

.10944 

.17083 

.17222 

.17500 

.17778 

.17917 

.  18050   10 

11 

.18333 

.18611 

.18750 

.18889 

.19167 

19444 

.19.583 

.19723   11 

12 

.20000 

.20278 

.20417 

.30556 

.20833 

.21111 

.21260 

.2KJS!)   !;> 

13 

.21667 

.21944 

.99081 

[22228 

.2251)0 

.22778 

.22917 

.2:l(  C><>   13 

14 

.23333 

.23611 

.88750 

.23889 

.24167 

.94444 

[34588 

.24722   14 

15 

.25000 

.25278 

.25417 

.25556 

.25833 

.26111 

.26250 

.2038!*   15 

16 

.26607 

.20914 

.27083 

.27332 

.37600 

.27778 

.27917 

.28050   10 

17 

.283:33 

.28611 

.28750 

.28889 

.21)167 

.29444 

.29588 

.29722   17 

18 

.30000 

.30278 

.30417 

.30556 

.30833 

.31111 

.31250 

.3K38!)   18 

19 

.31067 

.31944 

.32083 

.32232 

.82500 

.32778 

.32917 

..V.050   19 

20 

.33333 

.33011 

.33750 

.33889 

.34167 

.34444 

.34583 

.34722  20 

21 

.35000 

.35278 

.35417 

.35556 

.35833 

.36111 

.86860 

.36389  21 

22 

.86667 

.30944 

.37083 

.37222 

.37500 

.3777'8 

.37917 

3805(5   22 

23 

.38333 

.38011 

.38750 

.38889 

.89167 

.39444 

.39583 

.397  22   2:5 

24 

.40000 

.40278 

.40417   .10556 

.40833 

.41111 

.41250 

.41389  21 

25 

.41667 

.41944 

.42083 

.42232 

.42500 

.48778 

.42917 

.43056  25 

26 

.433:33 

.43611 

.43750 

.438S9 

.44167 

.44444 

.44588 

.44722 

20 

27 

.45000 

.45278 

.45417 

.45550 

.45833 

.46111 

.46250  !  .40389 

27 

28 

.46667 

.46944 

.47083 

.47328 

.47500 

.4777'8 

.471)17   .-1S056  28 

29 

.48*33 

.48611 

.48750 

.48889 

.49167 

.49441 

.49588 

.•ID:  22  29 

30 

.50000 

.50278 

.50417 

.5<J556 

.50833 

.51111 

.51250 

.5138!)  :»0 

31 

.51667 

.51944 

.52083 

.52222 

.52500 

.52778 

.E2917 

.53056  i  31 

32 

.53:333 

.53611 

.53750 

.53889 

.54167 

.54444 

.54588 

.54722   32 

33 

.55000 

.55278 

.55417 

.55556 

.55S.-53 

.50111 

.66250 

.fi63S!»   *} 

34 

.56667 

.56944 

.57083 

.57223 

.57500 

.57778 

.57917 

.58056  34 

35 

.58333 

.58611 

.58750 

.58889 

.59167 

.59444 

.59583 

.5»722   35 

36 

.60000 

.60278 

.60417 

!  60556 

.60833 

.61111 

.61260 

.til3H«>  36 

37 

.61667 

.61944 

.62083 

.63233 

.62500 

.62778 

.62917 

.03050   37 

38 

.03333 

.63611 

.63750 

.63889 

.64167 

.61111 

.64583 

.04722   38 

39 

.65000 

.65278 

.66417 

.65556 

.058:33 

.60111 

.66250 

.00:3X5)  39 

40 

.66667 

.66944 

.67083 

.67222 

.67500 

.07778 

.67917 

.68056 

40 

41 

.68333 

.68611 

.68750 

.68889 

.69167 

.69441 

.69583 

.69722 

41 

42 

.70000 

.70378 

.70417   .70556 

.71KU 

.71111 

.71250   .7138!)   12 

43 

.71667 

.71944 

.720S-!   .72222 

.73300 

178778 

.7:2017   .73050  43 

44 

.73333 

.73611 

.73750 

.73889 

.74167 

.74444 

.7458-') 

.71722   44 

45 

.75000   .75278 

.75417 

.75556 

.75833 

.70111 

.70250 

.7'f:'3S!>   45 

46 

.7r,!>(;7  .7'09i4 

.77083 

[77223 

.77500 

.77778 

.771117 

.78066 

46 

47 

.78333 

.78611   .78750 

.78889 

.79167 

.79444 

.79588 

[79728 

47 

48 

.80000 

.80378   .80417 

180556 

180888 

.81111 

.S  1250 

.81:38!)   is 

49 

.81667 

.S19U   .82083 

.83333 

.83500 

.88778 

[82917 

.S3056   -4!) 

50 

.83333 

.83611 

.83760 

.83889 

.84167 

.SI  111 

[84688 

.81722 

50 

51 

.85000 

.85278 

.85417 

.85556 

.85833 

86111 

.80250 

.86869 

51 

52 

.80607 

.86044 

.87083 

.87222 

[87500 

.87778 

.87917 

.88056 

68 

53 

.88333   .88611 

.887'50 

.88889 

.89167 

.89444 

89588 

.89728 

68 

54 

.90000  !  .90278 

.90417 

.90556 

.90888 

.91111 

91250 

.91889 

54 

55 

.91667 

.91944 

.92083 

[92222 

.92500 

.!)2778 

92917 

.93050   55 

56 

.93333 

.(Mill 

.93750 

.93889 

94167 

.94444 

94588 

.1)4722  j  56 

57 

.95000 

.86878 

.95417 

.95556 

95888 

.96111 

96290 

.9I53M)  57 

58 

.%<iC,7 

.06944 

.97088 

.97222   .{)75(iO 

.97778 

97917 

.98056 

58 

59 

.98333 

.98611 

.98750 

.98889  j 

.99167 

.99444 

99583 

.!)!)722 

59 

/ 

0" 

10" 

15" 

20" 

30" 

40" 

45" 

50" 

/ 

264. 


TABLE  VIII.— SQUARES,  CUBES,  SQUARE  ROOTS,  AND  CUBE  ROOTS. 


No. 

Squares. 

Cubes. 

Square 

Roots. 

Cube  Roots. 

"I 
Reciprocals. 

1 

1 

1 

1.0000000 

1.0000000 

1.000000000 

2 

4 

8 

1.4142136 

1.2599210 

.500000000 

3 

9 

27 

1.7320508 

1.4422496 

.333333:333 

4 

16 

64 

2.0000000 

1.5874011 

.250000000 

5 

25 

125 

2.  2360680 

1.7099759 

.200000000 

6 

36 

216 

2  4494897 

1.8171206 

.166666667 

7 

49 

343 

2.6457513 

1.9129312 

.142857143 

8 

64 

512 

2.8284271 

2.0000000 

.125000000 

(J 

81 

729 

3.0000000 

2.0800837 

.111111111 

10 

100 

1000 

3.1622777 

2.1544347 

.100000000 

11 

121 

1331 

3.3166248 

2.2239801 

.090909091 

12 

144 

1728 

3.4641016 

2.2894286 

.0833333:33 

13 

169 

2197 

3.6055513 

2.3513347 

.076923077 

14 

196 

2744 

3.7416574 

2.4101422 

.071428571 

15 

225 

3375 

3.8729833 

2.4662121 

.066666667 

16 

25(5 

4096 

4.0000000 

2.5198421 

.062500000 

17 

289 

4913 

4.1231056 

2.5712816 

.058823529 

18 

324 

5832 

4.2426407 

2.6207414 

.055555556 

19 

361 

6859 

4.3588989 

2.6684016 

.052631579 

20 

400 

8000 

4.4721360 

2.7144177 

.050000000 

21 

441 

9261 

4.5825757 

2.7589243 

.047619048 

00 

484 

10648 

4.6904158 

2.8020393 

.045454545 

23 

529 

12167 

4.7958315 

2.843867'Q 

.043478261 

24 

576 

13824 

4.8989795 

2.8844991 

.041666667 

25 

625 

15625 

5.0000000 

2.1)240177 

.040000000 

26 

676 

17576 

5.0990195 

2.11624960 

.038461538 

27 

729 

19683 

5.1961524 

3.0000000 

.037037037 

28 

784 

21952 

5.2915026 

3.0365889 

.035714286 

29 

841 

24:389 

5.3851648 

3.0723168 

.034482759 

30 

900 

27000 

5.4772256 

3.107'2325 

.033333333 

31 

961 

29791 

5.5677644 

3.1413806 

.032258065 

32 

1024 

32768 

5.6568542 

3.17'48021 

.031250000 

88 

1089 

35937 

5.7445626 

3.2075343 

030303030 

34 

1156 

39304 

5.8309519 

3.2396118 

.029411765 

35 

1225 

42875 

5.9160798 

3.2710663 

.028571429 

36 

1296 

46656 

6.0000000 

3.3019272 

.027777778 

37 

1369 

50653 

6.0827625 

3.3322218 

.027027027 

38 

1444 

54872 

6.1644140 

3.3619754 

.026315789 

39 

1521 

59319 

6.2449980 

3.3912114 

.025641026 

40 

1600 

64000 

6.3245553 

3.4199519 

.025000000 

41 

1681 

68921 

6.4031242 

3.4482172 

.024390244 

42 

1764 

74088 

6.4807407 

3.4760266 

.023809524 

43 

1849 

7i  ir.or 

6.5574385 

3.50*3981 

.023255814 

44 

1986 

85184 

6.6332496 

3.5303483 

.022727273 

45 

2025 

91135 

6.7082039 

3.5568933 

.022222222. 

46 

2116 

97336 

6.7823300 

3.5830479 

.021739130 

47 

2209 

103823 

6.8556546 

3.6088261 

.02127'6600 

48 

2304 

110592 

6.9282032 

?.  6342411 

.020833333 

49 

2401 

117649 

7.0000000 

'3.6593057 

.020408163 

50 
51 

2500 
2601 

125000 
132651 

7.0530678 

.  MU'.'S  I 

3.6840314 

3.7(184298 

.020000000 
.019607843 

52 

2704 

140608 

.2111026 

3.7325111 

.019230769 

53 

2809 

148877 

.2801099 

3.7562858 

.018867925 

54 

2916 

15?'464 

.3484692 

3.7797631 

.018518519 

55 

3025 

166375 

.4161985 

3.8029525 

.018181818 

56 

3136 

175616 

.4833148 

3.8258624 

.017857143 

57 

3249 

185193 

.5498344 

3.8485011 

.017543860 

58 

3364 

195112 

.6157731 

3.8708766 

.017241379 

59 

3481 

205379 

.6811457 

.  3.8929965 

.016949153 

60 

3600 

216000 

.7459667 

3.9148676 

.016666667 

61 

3721 

226981 

.8102497 

3.93(54972 

.016393443 

62 

3844 

238328 

7.8740079 

3.9578915 

.016129032 

265 


TABLE  vm.- continued. 


No.   Squares. 

Cubes. 

Scpiare 
Roots. 

Cube  Roots. 

Reciprocals. 

63 

3969 

250047 

7.9372539 

8.9700571 

.015873016 

64 

4096 

262144 

8.0000000 

4.0000000 

.015025000 

65 

4225 

274625 

8.0622577 

4.0207256 

.015384015 

66 

4356 

287496 

8.1240384 

4.0412401 

.015151515 

67 

4489 

300763     8.1853528 

4.0615480 

.014925373 

68 

4624 

314432 

8.2462113 

4.0816551 

.014705882 

69 

4761 

328509 

8.3066239 

4.1015661 

.014492754 

70' 

4900 

343000 

8.3666003 

4.1212853 

.014285714 

71 

5041 

357911 

8.4261498 

4.1408178 

.014084507 

72 

5184 

373248 

8.4852814 

4,1601676 

.01388K881) 

73 

5329 

389017 

8.5440037 

4.1793390 

.013698030 

74 

5476 

405224 

8.6023253 

4.198S364 

.013513514 

75 

5625 

421875 

8.6602540 

4.2171633 

.013333333 

76 

5776 

438976 

8.7177979 

4.2358236 

.013157895 

77 

5929     456533 

8.7749644 

4.2343210 

.012..87013 

78 

6081 

474552 

8.8317609 

4.2726586 

.012820513 

79 

6241 

493039 

8.8881944 

4.2908404 

.012058228 

80 

6400 

512000 

8.9442719 

4.3088695 

.012500000 

81 

6561 

531441 

9.0000000 

4.3267487 

.012345679 

82 

6724 

551368 

9.0553851 

4.3441815 

.012195122 

83 

6889 

571787 

9.1104336 

4.3020707 

.012048193 

84 

7056 

592704 

9.1651514 

4.3795191 

.011904702 

85 

7225 

614125 

9.2195445 

4.3968296 

.011764706 

86 

7396 

636056 

9.273G185 

4.4140049 

.011027907 

87 

7569 

658503 

9.3273791 

4.4310476 

.01  149  1-253 

88 

7744 

681472 

9.3808315 

4.4479602 

.011303(530 

89 

7921 

704969 

9.4339811 

4.4647451 

.011235955 

90 

8100 

729000 

9.4868330 

4.4814047 

.011111111 

91 

8281 

753571 

9.5393920 

4.4979414 

.010989011 

92 

8464 

778688 

9.5916630 

4.5143574 

.010869565 

93 

8649 

804357 

9.6430508 

4.5306549 

.010752688 

94 

8836 

83038-4 

9.6953597 

4.5468359 

.010638298 

95 

9025 

857375 

9.7467U43 

4.5629026 

.010520316 

96 

9216 

884736 

9.7979590 

4.5788570 

.010416607 

97 

9409 

912673 

9.8488578 

4.5947009 

.010309278 

98 

9604 

941192 

9.8994949 

4.6104303 

.010204082 

99 

9801 

970299 

9.9498744 

4.6260650 

.010101010 

100 

10000 

1000000 

10.0000000 

4.6415888 

.010000000 

101 

10201 

1030301 

10.0498756 

4.657W95 

.009900990 

102 

10404 

1061208 

10.0995049 

4.6723287 

.009803922 

103 

10609 

1092727 

10.1488916 

4.6875482 

.009708738 

104 

10816 

1124864 

10.1980390 

4.  70206!  14 

.009615385 

105 

11025 

1157625 

10.2469508 

4.7176940 

.009523810 

106 

11236 

1191016 

10.2956301 

4.7320235 

.009433962 

107 

11449 

1225043 

10.3440804 

4.7474594 

.009345794 

108 

11604 

1259712 

10.3923048 

4.7622032 

.00  -259259 

109 

11881 

1295029 

10.4403065 

4.7768562 

.009174313 

110 

12100 

3331000 

10.4880885 

4.7914199 

.009000009 

111 

12321 

1367631 

10.5:556538 

4.8058955 

.609009009 

112 

12544 

1404928 

10.5860052 

4.8202845 

[0089285.  1 

113 

12769 

1442897 

10  !  6301  458 

4.8345881 

.008811(558 

114 

12996 

1481544 

10.  (770783 

4.8488076 

.008771930 

115 

13225 

1520875 

10.7238058 

4.8629442 

.008605052 

116 

13456 

1560896 

10.7703296 

4.8700000 

.008020690 

117 

13689 

1601613 

10.8166538 

4.890»7'.'fcJ 

.008547009 

118 

18924 

1643083 

10.8627805 

4.9048681 

.008474576 

119 

14161 

1685159 

10.9087121 

4.9186847 

.008403361 

120 

14400 

1728000 

10.9544512 

4.  93242  IS 

.008:333333 

121 

14641 

1771561 

11.00  0000 

4.9400874 

.008264463 

m 

14884 

1815848 

11.0453610 

4.9396757 

.  008100  721 

123 

18129 

1S<;0867 

11.0005865 

4.  (1731898 

.008130081 

124     15:576 

190IKW  I 

11.1.335287 

4.9800310 

.008004516 

260 


TABLE  Vlll.- Continued. 


No. 

Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots. 

Reciprocals. 

125 

15625 

1953125 

11.1803399 

5.0000000 

.008000000 

126 

15876 

2000376 

11.2249722 

5.0132979 

.007936508 

127 

16129 

2048383 

11.2694277 

5.0265257 

.007874016 

128 

16384 

2097152 

11.3137085 

5.0396842 

.007812500 

129 

16641 

2146689 

11.3578167 

5.0527743 

.007751938 

130 

16900 

2197000 

11.4017543 

5.0657970 

.007692308 

131    17161 

2248091 

11.4455231 

5.0787531 

.007633588 

132    17424 

2299968 

11.4891253 

5.0916434 

.  007575758 

133 

17689 

2352637 

11.5325636 

5.1044687 

.007518797 

134 

17956 

2406104 

11.5758369 

5.1172299 

.007462687 

135 

18225 

2460375 

11.6189500 

5.1299278 

.007407407 

136 

18496 

2515456 

11.6619038 

5.1425632 

.007352941 

137 

18769 

2571353 

11.7046999 

5.1551367 

.007299270 

138 

19044 

2628072 

11.7473401 

5.1676493 

.007246377 

139 

19321 

2685619 

11.7898261 

5.1801015 

.007194245 

140 

19600 

2744000 

11.8321596 

5.1924941 

.007142857 

141 

19881 

2803221 

11.8743421 

5.2048279 

.007092199 

142 

20164 

2863288 

11.9163753 

5.2171034 

.007042254 

143 

20449 

2924207 

11.9582607 

5.2293215 

.006993007 

144 

20736 

2985984 

12.0000000 

5.2414828 

.006944444 

145 

21025 

3048625 

12.0415946 

5.2535879 

.006896552 

146    21316 

3112136 

12.0830460 

6.2656374 

.006849315 

147  i   21609 

3176523 

12.1243557 

5.2776321 

.006802721 

148 

21904 

3241792 

12.1655251 

5.2895725 

.006756757 

149 

22201 

3307949 

12.2065556 

5.3014592 

.006711409 

150 

22500 

3375000 

12.2474487 

5.3132928 

.006666667 

151 

22801 

3442951 

12.2882057 

5.3250740 

.006622517 

152 

23104 

3511808 

12.3288280 

5.3368033 

.006578947 

153 

23409 

3581577 

12.3693169 

5.3484812 

.006535948 

154 

23716 

3652264 

12.4096736 

5.3601084 

.006493506 

155 

24025 

3723875 

12.4498996 

5.3716854 

.006451613 

156 

24336 

3796416 

12.4899960 

5.3832126 

.006410256 

157 

24649 

3869893 

12.5299641 

5  '3946907 

.006369427 

158 

24964 

3944312 

12.5698051 

5.4001202 

.006329114 

159 

25281 

4019679 

12.6095202 

5.4175015 

.006289308 

160 

25600 

4096000 

12.6491106 

5.4288352 

.006250000 

161 

25921 

417'3281 

12.6885775 

5.4401218 

.C06211180 

162 

26244 

4251528 

12.7279221 

5.4513618 

.00617'2840 

163 

26569 

4330747 

12.7671453 

5.4625556 

.006134969 

164 

26896 

4410944 

12.8062485 

5.4737037 

.006097561 

165 

27225 

4492125 

12.8452326 

5.4848066 

.006060606 

166 

27556 

4574296 

12.8840987 

5.4958647 

.006024096 

167 

27889 

4657463 

12.9228480 

6.50G8784 

.005988024 

168 

28224 

4741632 

12.9614814 

5.  5178484 

.005952381 

169 

28561 

4826809 

13.0000000 

5.5287748 

.005917160 

170 

28900 

4913000 

13.0384048 

5.5396583 

.005882353 

171 

29241 

5000211 

13.0766968 

5.5504991 

.005847953 

172 

29584 

5088448 

13.1148770 

,0.5612978 

.005813953 

173 

29929 

5177717 

13.1529464 

5.5720546 

.005780347 

174 

30276 

5268024 

13.1909060 

5.5827702 

.005747126 

175 

30625 

5359375 

13.2287566 

5.51)34447 

.005714286 

176 

30976 

5451776 

13.2664992 

5.6040787 

.005681818 

177 

31329 

5545333 

13.3041347 

5.6146724 

.005649718 

178 

31684 

5639752 

13.3416041 

6.6252263 

.005617978 

179 

32041 

5735339 

13.3790882 

5.6357408 

.005586592 

180 

32400 

5832000 

13.4164079 

5.6462162 

.005555556 

181 

32761 

5929741 

13.4536240 

5.6566528 

.005524862 

182 

33124 

6028568 

13.4907376 

5.6670511 

.005494503 

183 

33489 

6128487 

13.5277493 

5  6774114 

.005464481 

184 

&S856 

6229504 

13.5646(500 

5.6877340 

.005434783 

185 

34225 

6331625 

13.6014705 

5.6980192 

.005405405 

186  |   34596 

6434856 

13.6381817 

5.7082675 

.005376344 

26? 


TABLE  VJK.— Continued. 


No.   Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots.  Reciprocals. 

187 

34969 

6539203 

13.6747943 

5.7184791 

.005347594 

188 

35344 

6644672 

13.7113092 

5.7286543 

.005319149 

189 

$572! 

6751269 

13.7477271 

5.7387936 

.005291005 

190 

36100 

6859000 

13.7840488 

5.7488971 

.005263158 

191 

36481 

6967871 

13.8202750 

5.7589652      .005235602 

192 

36864 

7077888 

13.8564065 

5.7689982      .005208333 

193 

37249 

7189057 

13.8924440 

5.7789966 

.005181347 

194 

37636 

7301384 

13.9283883 

5.7889604 

.005154639 

195 

38025 

7414875 

13.9642400 

5.7988900 

.005128205 

196 

38416 

7529536 

14.0000000 

5.8087857 

.005102041 

197 

38809 

7645373 

14.0356688 

5.8186479 

.005076142 

198 

39204 

7762392 

14.0712473 

5.8284767 

.005050505 

199    39601 

7880599 

14.1067360 

5.8382725 

.005025126 

200    40000 

8000000     14.1421356 

5.8480355 

.005000000 

201 

40401 

8120601      14.1774469 

5.8577660 

.004975124 

202 

40804 

8242408      14.2126704 

5.8674643 

.004950495 

203 

41209 

8365427     14.2478068 

5.8771307 

.004926108 

204 

41616 

8489664     14.2828569 

5.8867653 

.004901961 

205     42025 

8615125     14.3178211 

5  .8963685 

.004878049 

206     42436 

8741816     14.3527001 

5.9059406 

.004854369 

207 

42849 

8869743     14.3874946     5.9154817 

.004830918 

208     43264 

8998912     14.4222051     5.9249921 

.004807692 

209     43681 

9129329 

14.4568323  ;   5.9344721 

.004784689 

210    44100 

9261000 

14.4913767 

5.9439220 

.004761905 

211     44521 

9393931 

14.5258390 

5.9533418 

.004739336 

212  !   44944 

9528128 

14.5602198 

5.9627320 

.004716981 

213     45369 

9663597 

14.5945195 

6.9720926 

.004694836 

214     45796 

9800344 

14.6287388     5.9814240 

.004672897 

215 

46225 

9938375 

14.6628783  i   5.  9907264 

.004651163 

216 

46656 

10077696 

14.6969385 

6.0000000 

.004629630 

217 

47089 

10218313 

14.7309199 

6.0092450 

.004608295 

218 

47524 

10360232 

14.7648231 

6.0184617 

.004587156 

219 

47961 

10503459 

14.7986486 

6.0276502 

.004566210 

220 

48400 

10648000 

14.832397'0 

6.0368107 

.004545455 

221 

48841 

10793861 

14.8660687 

6.0459435 

.004524887 

222 

49284 

10941048 

14.8996644 

6.0550489 

.004504505 

223 

49729 

11089567 

14.9331845 

6.0641270 

.0044H4305 

224 

50176 

11239424 

14.9666295 

6.0731779 

.004464286 

225 

50625 

11390625 

15.0000000 

6.C822020 

.004444444 

226 

51076 

11543176 

15.0332964 

6.0911994 

.004424779 

227 

51529 

11697083 

15.0665192 

6.1001702 

.004405286 

228 

51984 

11852352 

15.0996689 

6.1091147 

.004383965 

229 

52441 

12008989 

15.1327460 

6.1180332 

.004366812 

230 

52900 

12167000 

15.1657509 

6.1269257 

.004347'826 

231 

53361 

12326391 

15.1986842 

6.1357924 

.004329004 

232 

53824 

12487168 

15.2315462 

6.1446337 

.004310345 

233 

54289 

12649337 

15.2643375 

6.1534495 

.004291845 

234 

54756 

12812904 

15.2970585 

6.1622401 

.004273504 

235 

55225 

12977875 

15.3297097 

6.1710058 

]  004255319 

236 

55696 

13144256 

15.3622915 

6.1797'466 

.004237288 

237 

56169 

13312053 

15.3948043 

6.1884628 

.004219409 

238 

56644 

13481272 

15.4272486 

6.1971544 

.004201681 

239 

57121 

13651919 

15.4596248 

6.2058218 

.004184100 

240 

57600 

13824000 

15.4919334 

6.2144650 

.004166667 

241 

58081 

13997521      15.5841747 

6.2230843 

.004149378 

242 

58564 

14172488     15.5563492 

6.2316797 

.004132231 

243 

59049 

14348907     15.5884573 

6.2402515 

.004115226 

244 

59536 

14526784     15.6204994 

6.2487'!)98 

.004098361 

245 

60025 

14706125     15.6524758 

6.2573248 

.004081633 

246 

60516 

14886936     15.6843871 

6.2658266 

.004065041 

247 

61009 

15069223     15.7162336 

6.2743054 

.004048583 

248 

61504 

15252992     15.7480157 

6.2827613 

.004032258 

TABLE  VUI.-Continued. 


I 

No.    Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots. 

Reciprocals. 

249 

62001 

15438249 

15.7797338 

6.2911946 

.004016064 

250 

62500 

15625000 

15.8113883 

6.2996053 

.004000000 

251 

63001 

15813251 

15.8429795 

6.3079935 

.003984004 

252 

63504 

16003008 

15.8745079 

6.3163596 

.003968254 

253     64009 

16194277 

15.9059737 

6.3247035 

.003952569 

254 

64516 

16:387064 

15.9373775 

6.3330256 

.003937008 

255 

65025 

16581375 

15.9687194 

6.3413257 

.003921569 

256 

65536 

16777216 

16.0000000 

6.3496042 

.003906250 

257 

66049 

16974593 

16.0312195 

6.3578611 

.003891051 

258 

66564 

17173512 

16.0623784 

6.3660968 

.00:3875969 

259 

67081 

17373979 

16.0934769 

6.3743111 

.00:3861004 

260 

67600 

17576000 

16.1245155 

6.3825043 

.003846154 

261 

68121 

17779581 

16.1554944 

6.3906765 

.003831418 

262 

68644 

17984728 

16.1864141 

6.3988279 

.003816794 

263 

69169 

18191447 

16.2172747 

6.4069585 

.00:3802281 

264 

69696 

18399744 

16.2480768 

6.4150687 

.003787879 

265 

70225 

18609625 

16.2788206 

6.4231583 

.003773585 

266 

70756 

18821096 

16.3095064 

6.4312276 

.003759398 

267 

71289 

190:34163 

16.3401346 

6.4392767 

.003745318 

268 

71824 

19248832 

16.3707055 

6.4473057 

.003731343 

269 

72361 

19465109 

16.4012195 

6.4553148 

.003717472 

270 

72900 

19683000 

16.4316767 

6.4633041 

.003703704 

271 

73441 

19902511 

16.4620776 

6.4712736 

.003690037 

272 

73984 

20123618 

16.4924225 

6.4792236 

.00-3676471 

273 

7152;) 

20346417 

16.5227116 

6.4871541 

.003663004 

274 

75076 

20570824 

16.5529454 

6.4950653 

.00:3649635 

275 

75625 

20796875 

16.5831240 

6.5029572 

.003636364 

276 

76176 

21024576 

16.6132477 

6.5108300 

.003623188 

277 

76729 

21253933 

16.6433170 

6.5186839 

.003610108 

278 

77284 

21484952 

16.6733320 

6.5265189 

.003597122 

279 

77841 

21717639 

16.7032931 

6.5343351 

.003584229 

280 

78400 

21952000 

16.7332005 

-6.5421326 

.00a571429 

281     78961 

22188041 

16.7630546 

6.5499116 

.003558719 

282 

79524 

22425768 

16.7928556 

6.5576722 

.003546099 

283 

80089 

22665187 

16.8226038 

6.5654144 

.003533569 

284 

80656 

22906:504 

16.8522995 

6.5731385 

.00:3521127 

285 

81225 

23149125 

16.8819430 

6.5808443 

.003508772 

286 

81796 

23393656 

16.9115345 

6.5885323 

.003496503 

287 

82369 

23639903 

16.9410743 

6.5962023 

.003484321 

288 

82944 

23887872 

16.9705627 

6.6038545 

.003472222 

289 

83521 

24137569 

17.0000000 

6.6114890 

.003460208 

290 

84100 

24389000 

17.0293864 

6.6191060 

.003448276 

291 

84681 

24642171 

17.0587221 

6.6267054 

.003436426 

292 

85264 

24897088 

17.0880075 

6.6342874 

.003424658 

293 

85849 

25153757 

17.1172428 

6.6418522 

.003412969 

294 

86436 

25412184 

17.1464282 

6.6493998 

.00:3401361 

295 

87025 

25672375 

17.1755640 

6.6569302 

.003389831 

296 

87616     25934a36 

17.2046505 

,  6.6644437 

.003378378 

297 

88209 

26198073 

17.2336879 

6.6719403 

.003367003 

298 

88804 

26463593 

17.2626765 

6.6794200 

.003355705 

299 

89401 

26730899 

17.2916165 

6.6868831 

.003344482 

300 

90000 

27000000 

17.3205081 

6.6943295 

.003333333 

301 

90601 

27270901 

17.3493516 

6.7017593 

.003322259 

302 

91204 

.27543608 

17.3781472 

6  7091729 

.003311258 

303 

91809 

27818127 

17.4068952 

6.7165700 

.003300330 

304 

92116 

28094464 

17.4355958 

6.7239508 

.003289474 

305 

93025 

28372625 

17.4642492 

6.7313155 

.003278689 

306 

93636 

28652616 

17.4928557 

6.7386641 

.003267974 

307     94249 

28934443 

17.5214155 

6.7459967 

.003257329 

308 

94864     29218112 

17.5499288 

6.7533134 

.003246753 

309 

95481 

29503629     17.5783958     6.7606143 

.003236246 

310 

96100 

29791000    17.6068169    6.7678995 

.003225806 

269 


TABLE  VIII.— Continued. 


1 

1 

No. 

Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots,  i 

Reciprocals. 

311 

96721 

30080231 

17.6351921 

6.7751690 

.003215434 

312 

97344 

30371328 

17.0035217 

6.7824229 

.003205128 

313 

97969 

30604297 

17.6918000 

6.7896613 

.003194888 

314 

98596 

30959144 

17.7200451 

6.7908844 

.003184713 

315 

99225 

31255875 

17.7482393 

6.8040921 

.003174603 

316 

99856 

31554496 

17.7703888 

6.8112847 

.003104557 

317 

100489 

31855013 

17.8044938 

6.8184020 

.003154574 

318 

101124 

32157432 

17.8325545 

6.8250242 

.003144054 

319 

101761 

32461759 

17.8005711 

0.8327714 

.003134790 

320 

102400 

32708000 

17.8885438 

6.8399037 

.003125000 

321 

103041 

33076161 

17.U104729 

'6.8470213 

.003115205 

322 

103684 

33386248 

17.9443584 

6.8541240 

.003105590 

323 

104329 

33698267 

17.97'22()08 

6.8012120 

.003095975 

324 

104976 

34012224 

18.0000000 

6.8082855 

.003080420 

325 

105625 

34328125 

18.0277504 

6.8753443 

.003076923 

320 

106276 

3464597'6 

18.05547W 

6.8823888 

.003067485 

327 

106929 

349657B3 

18.0831413 

6.8894188 

.003058104 

328 

107584 

35287552. 

18.1107703 

6.8964345 

.003048780 

329 

108241 

35011289 

18.1383571 

6.9034359 

.003039514 

330 

108900 

35937000 

18.1659021 

6.9104232 

.003030303 

331 

109501 

36204091 

18.1934054 

6.9173904 

.003021148 

332 

110224 

30594368 

18.2208072 

6.9243556 

.003012048 

333 

110889 

36U26037 

18.2482876 

6.9313008 

.003003003 

334 

111556 

37259704 

18.2756069 

6.9382321 

.002994012 

335 

112225 

37595375 

18.3030052 

6.9451496 

.002985075 

336 

112896 

37933056 

18.3303028 

6.9520533 

.002976190 

337 

113569 

38272753 

18.3575598 

6.9589434 

.002967359 

338 

114244 

3861447'2 

18.3847703 

6.9658198 

.002958580 

339 

114921 

38958219 

18.4119526 

6.9726826 

.002949853 

340 

115600 

39304000 

18.4390889 

6.9795321 

.002941176 

341 

116281 

39651821 

18.4061853 

6.9863081 

.002932551 

342 

116964 

40001688 

18.4932420 

6.9931906 

.002923977 

343 

117649 

40353607 

18.5202592 

7.0000000 

.002915452 

344 

118336 

40707584 

18.547237'0 

7.0067962 

.002906977 

345 

119025 

41063025 

18.57'417'50 

7  0135791 

.002898551 

346 

119716 

41421736 

18.6010752 

7.0203490 

.002890173 

347 

120409 

41781923 

18.627-9360 

7.0271058 

.002881844 

348 

121104 

42144192 

18.6547'581 

7.03:38497 

.002873503 

349 

121801 

42508549 

18.6815417 

7.0405806 

.002865330 

350 

122500 

42875000 

18.7'082869 

7.0472987 

.002857143 

351 

123201 

43243551 

18.7:349940 

7.0540041 

.002849003 

352 

123904 

43614208 

18.70160:30 

7.0000967 

.002840909 

353 

124609 

43986977 

18.7882942 

7.0673767 

.002832861 

354 

125316 

44361864 

18.8148877 

7.0740440 

.002824859 

355 

126025 

44738875 

18.8414437 

7.0800988 

.002816901 

356 

126736 

45118016 

18.8079023 

7.0873411 

.002808989 

357 

127449 

45499293 

18.8944436 

7.0939709 

.002801120 

358 

128164 

45882712 

18.9208879 

7.1005885 

.002793296 

359 

128881 

46268279 

18.9472953 

7.1071937 

.002785515 

360 

129600 

46656000 

18.9730600 

7.1137806 

.002777778 

361 

130321 

47'045881 

19.0000000 

7.1203674 

.002770083 

362 

131044 

47437928 

19.0262976 

7.1269360 

.0027-62431 

363 

131769 

47832147 

19.0525589 

7.1334925 

.002754821 

364 

132496 

48228544 

19.0787840 

7.1400370 

.002747253 

365 

133225 

48627125 

19.1049732 

7.1405695 

.002739726 

366 

133956 

49027896 

19.1311265 

7.15:30901 

.0027-32240 

367 

134089 

49430863 

.19.1572441 

7.1595988 

.002724796 

368 

135424 

498:36032 

19.18:33261 

7.ioooa-)7 

.002717391 

369 

130161 

50243409 

19.2093727 

7.17'25809 

.002710027 

370 

136900 

50653000 

19.2353841 

7.1790544 

.002702703 

371 

137641 

51004811 

19.2(>i::r,o:3 

7.1855162 

.002695418 

372 

138384 

51478848 

19.2873015 

7.1919663 

•  .002688172 

270 


TABLE  VIII.— Continued. 


No. 

Squares. 

Cubes. 

Square 
Hoots. 

Cube  Roots. 

Reciprocals. 

373 

139129 

51895117     19.3132079     7.1984050 

.002680965 

374 

139876 

52313624  |   19.3390796  !   7.2048322 

.002673797 

375 

140625 

52734375 

19.3649167     7.2112479 

.002666667 

376 

141376 

53157376 

19.3907194     7.2176522 

.002659574 

377 

142129 

53582033 

19.4164878     7.2240450 

.002652520 

378 

1428*4 

54010152  !   19.4422221     7.2304268 

.002645503 

379 

143641 

54439939 

19.4079^23     7.2367972 

.002638522 

380 

144100 

54872000 

19.4935887  :   7.2431565 

.002631579 

381 

145161 

55306341 

19.5192213 

7.2495045 

.002024672 

38-3 

145924 

55742908 

19.5448203 

7.2558415 

.002617801 

38.5 

14(5089 

50181887 

19.5703858 

7.2621675 

.002610966 

384 

147456 

50023104 

19.5959179     7.2684824 

.002004107 

385 

148225 

57000025 

19.6214169  !   7.2747804 

.002597403 

386 

148996 

57512456 

19.6468827 

7.2810794 

.002590074 

387 

149709 

57900003 

19.0723156 

7.2873617 

.002583979 

388 

150541 

58411072 

19.6977156 

7.2936330 

.002577320 

389 

151321 

5886:3809 

19.7230829 

7.2998936 

.002570694 

390 

152100 

59319000 

19.7484177 

7.3061436 

.002564103 

391 

152881 

59776471 

lit.  7  737199 

7.3123828 

.002557545 

392    153004 

60236288 

19.7989899 

7.3180114 

.002551020 

393 

154449 

60098457 

19.8242276 

7.3248295 

.002544529 

394 

155236 

61162984 

19.8494332 

7.3310369 

.002538071 

395 

156025 

61629875 

19.8746009 

7.337'2339 

.002531646 

390 

156816 

62099136 

19.8997487 

7.3434205 

.00x525253 

397 

157609 

62570773 

19.9248588 

7.3495900 

.002518892 

398 

158404 

63044792 

19.9499373 

7.3557024 

.002512563 

399 

159201 

63521199 

19.9749844 

7.3619178 

.0025U0206 

400 

160000 

64000000 

20.0000000 

7.3680630 

.002500000 

401 

160801 

64481201 

20.0249844  i   7.3741979 

.002493766 

402 

161604 

64964808 

20.0499377 

7.3803227 

.002487562 

403 

162409 

65450827 

20.0748599 

7.3864373 

.002481390 

404 

163216 

05939264 

20.0997512     7.3925418 

.002475248 

405 

164025 

60430125 

20.1246118     7.3986363 

.002409136 

400 

164836 

60923416 

20.1494417     7.4047206 

.002403054 

407 

165649 

67419143 

20.1742410     7.4107950 

.002457002 

408 

160404 

67917312 

20.1990099  !   7.4168595 

.002450980 

409 

167281 

68417929 

20.2237484 

7.4229142 

.002444988 

410 

168100 

68921000 

20.2484567 

7.4289589 

.002439024 

411 

168921 

69426531 

20.2731349 

7.4849938 

.002433090 

412 

169744 

69934528 

20.2977831 

7.4410189 

.002427184 

413 

170509 

70444997 

20.3224014 

7.4470342 

.002421308 

414 

171396 

70957944 

20.3469899 

7.4530399 

.002415459 

415 

172225 

71473375    20.3715488 

7.4590359 

.002409639 

416 

173056 

71991296 

20.3960781 

7.4650223 

.002403846 

417 

173889 

72511713 

20.4205779 

7.4709991 

.002398082 

418 

174724 

73034632 

20.4450483 

7.4769664 

.002392344 

419 

175561 

73560059 

20.4694895 

7.4829242 

.002386635 

420 

176400 

74088000 

20.4939015 

7.4888724 

.002380952 

421 

177241 

74618461 

20.5182845 

7.4948113 

.002375297 

422 

178084 

75151448 

20.5426386 

7.5007406 

.002369668 

423 

178929 

75686967 

20.5669638 

7.5066607 

.002364066 

424 

179776 

76225024 

20  5912003 

7.5125715 

.002358491 

425 

180625 

70765625 

20.6155281 

7.5184730 

.002352941 

426 

181476 

77308776 

20.6397674 

7.5243652 

.002347418 

427 

182329 

77854483 

20.6639783 

7.5302482 

.002341920 

428 

183184 

78402752 

20.6881609 

7.5361221 

002336449 

429 

184041 

7S953589 

20.7123152 

7.5419867 

.002331002 

430 

184900 

79507000 

20.7364414 

7.5479423 

.002325581 

431 

185761 

80062991 

20.7605395 

7.5536888 

.002320186 

432 

186624 

80621508 

20.7846097 

7.5595263 

.00*314815 

433 

187489  1   811827R7 

20.8086520 

7  5053548 

.002309469 

434 

188:356  !   81740504 

20.a326667 

7.5711743 

.002304147 

271 


TABLE  VIII.— Continued. 


No. 

Squares. 

Cubes. 

Square 
Hoots. 

Cube  Roots. 

Reciprocals. 

435 

189225 

82312875 

20.8566536 

7.5769849 

.002298851 

436 

190096 

82881856 

20.88061:30 

7.5827865 

.00229^578 

437 

190969 

83453453 

20.9045450 

7.5885793 

.0022->8330 

438 

191844 

84027672 

20.9284495 

7.5943033 

.002283105 

439 

192721 

84604519 

20.9523208 

7.6001385 

.002277904 

440 

193600 

85184000 

20.9761770 

7.6059049 

.00227'27'27 

441 

194481 

85766121 

21.0000000 

7.6116026 

.002267574 

443 

195364 

86350888 

21.0237900 

7.6174116 

.002262413 

443 

196249 

86938307 

21.0475053 

7.6231519 

.002257336 

444 

197136 

87528384 

21.0713075 

7.6288837 

.002252252 

445 

198025 

88121125 

21.0950231 

7.  6346067 

.002247191 

446 

198916 

88716536 

21.1187121 

7.6403213 

.00:2242152 

447 

399809 

89314023 

21.1423745 

7.6460272 

.00223713(5 

448 

200704 

89915392 

21.1660105 

7.6517247 

.002232143 

449 

201601 

90518849 

21.1896201 

7.657'4133 

.002227171 

450 

202500 

91125000 

21.2132034 

7.6630943 

.002222222 

451 

203401 

9173:3851 

21.2367006 

7'.G(587'665 

.002217295 

452 

204304 

92345408 

21.2602916 

7.6744303 

.002212389 

453 

205209 

92959677 

21.2837967 

7.6800857 

.002207506 

454 

206116  i   93576664 

21.3072758 

7.6857328 

.0022021543 

455 

207025 

94196375 

21.8307290 

7.6913717 

.002197802 

456 

207938 

94818816 

21.a541505 

7.6970023 

.002192982 

457 

208849 

95443993 

21.3775583 

7.7026246 

.002188184 

458 

209764 

96071912 

21.4009346 

7.7082388 

.002183406 

459 

210681 

96702579 

21.4242853 

7.7138448 

.002178649 

460 

211600 

97336000 

21.4476106 

7.7194426 

.002173913 

461 

212521 

97972181 

21.4709106 

7.7'250325 

.002169197 

462 

213444 

98611128 

21.4941853 

7.7306141 

.002164502 

463 

214369 

99252847 

21.5174348 

7.7361877 

.002159827 

464 

215296 

99897344 

21.5406592 

7.7417'532 

.002155172 

465 

216225 

100544625 

21.5638587 

7.7473109 

.002150538 

466 

217156 

101194696 

21.5870331 

7.7528606 

.002145923 

467 

218089 

101847563 

21.6101828 

7.7584023 

.002141328 

468 

219024 

102503232 

21.6333077 

7.7639361 

.002136752 

469 

219961 

103161709 

21.6564078 

7.7694620 

.002132196 

470 

220900 

103823000 

21.6794834 

7.7749801 

.002127660 

471 

221841 

104487111 

21.7025344 

7.7'804904 

.00212:3140 

472 

222784 

105154048 

21.7255610 

7.7859928 

.0021  ISO  U 

473 

223729 

105823817 

21.7485632 

7.7914875 

.002114165 

474 

224676 

106496424 

21.7715411 

7.7969745 

.002109705 

475 

225625 

107171875 

21.7944947 

7.8024538 

.002105263 

476 

226576 

107850176 

21.8174242 

7.8079254 

.002100840 

477 

227529 

108531333 

21.8403297 

7.8133892 

.002096436 

478 

228484 

109215352 

21  8632111 

7.8188450 

.002092050 

479 

229441 

109902239 

21.8860686 

7.8242942 

.002087683 

480 

230400 

110592000 

21.9089023 

7.8297353 

.002088333 

481 

231361 

111284641 

21.9317122 

7.8351688 

.002079002 

482 

232324 

111980168 

21.9544984 

7.8405949 

.002074<»S!) 

483 

233289 

112678587 

21.9772610 

7.8460134 

.002070393 

484 

234256    113379904 

22.0000000 

7.8514244 

.0020(56110 

485 

235225    114084125 

22.0227155 

7.H568281 

.002061856 

486 

236196    114791256 

22.0454077 

7.8622242 

.002057613 

487 

237169    115501303 

22.0680765 

7.8676130 

,002058888 

488 

238144    11621427'2 

22.0907220 

7.87'29944 

.002049180 

489 

239121    116930169 

22.11*3444 

7.8783684 

.002044990 

490 

240100  i  117649000 

22.1359436 

7.8837352 

.002040810 

491 

241081 

118370771 

22.158515)8 

7.8890946 

.0020:30(5(50 

492 

242064 

119095488 

22.18107:50 

7.8944468 

.002032520 

493 

243049 

119823157 

2:2.2036033 

7.8907017 

.002028398 

494 

244036 

120553784 

22.22C.110S 

7.9051294' 

.002024291 

495 

245025 

121287375 

22.24859r>5 

7.9104599 

.002020202 

496 

246016 

12202:3!W6 

22.2710575 

7.9157832 

.002016129 

TABLE  VIII.— Continued. 


Xo. 

Squares. 

Cubes. 

Square 
Koots. 

Cube  Boots. 

Reciprocals. 

497 

347009 

133763473 

22.2934968 

-.  9210994 

.002012072 

498 

21S004 

133505'.)!  13 

22.3159136 

-.9264085 

.002008032 

499 

249001 

134351499 

22.3383079 

7.9317104 

.003004008 

500 

350000 

125000000 

22.3606798 

7.9370053 

.003000000 

501 

351001 

125751501 

22.3830293 

-.9422931 

.001996008 

503 

253.  i04 

130506008 

22.4053565 

7.9475739 

.001993033 

503 

353009 

13:303527 

22.4376615 

-.9528477 

.001988073 

504 

354016 

128024064 

22.4499443 

7.9581144 

.001984127 

505 

255035 

128787625 

22.4722051 

7.96a3743 

.001980198 

506 

356036 

129554216 

22.4944438 

7.9686271 

.001976285 

sor 

357049 

130323843 

22.5166605 

7.9738731 

.001972:387 

508 

258064 

131096512 

22.5!388553 

7.9791122 

.001968504 

509 

259081 

131872239 

22.5610283 

7.9843444 

.001964637 

510 

260100 

132651000 

22.5831796 

7.9895697 

.001960784 

511 

261121 

133432831 

22.6053091 

7.9947883 

.001956947 

513 

263144 

134217728 

22.  6274170 

8.0000000 

.001953125 

513 

263169 

135005697 

22.6495033 

8.0052049 

.001949318 

514 

204  196 

135796744 

22.6715681 

8.0104032 

.001945525 

515 

205335 

13659087'5 

22.6936114 

8.0155946 

.001941748 

516 

260356 

137388096 

22.7156334 

8.0207794 

.001937984 

517 

267289 

138188413 

22.7376340 

8.0259574 

.001934236 

518 

268324 

138991832 

22.7'596134 

8.0311287 

.001930502 

519 

269361 

139798359 

22.7815715 

8.0362935 

.001936782 

530 

270400 

140608000 

22.8035085 

8.0414515 

.001933077 

521 

271441 

141420761 

22.8254244 

8.0466030 

.001919386 

533 

273484 

142236648 

22.8473193 

8.0517479 

.001915709 

533 

273539 

143055667 

22.8691933 

8.0568862 

.001913046 

534 

274576 

143877824 

22.8910463 

8.0620180 

.001908397 

535 

275625 

144703125 

22.9128785 

8.0671432 

.001904763 

536 

276676 

145531576 

22.9346899 

8.07'22620 

.001901141 

537 

277729 

146363183 

22.9564806 

8.0773743 

.001897533 

538 

278784 

147197952 

22.9783506 

8.0R24800 

.001893939 

529 

279841 

148035889 

23.0000000 

8.0875794 

.001890359 

530 

280900 

148877000 

23.0217289 

8.0936733 

.001886792 

531 

281961 

1497'21291 

23.043437'2 

8.0977589 

.001883339 

533 

283024 

150568768 

23.0651252 

8.1038390 

.001879699 

533 

284089 

151419437 

23.0867928 

8.1079138 

.001876173 

534 

285156 

152273304 

23.1084400 

8.1139803 

.001872659 

535 

286225 

153130375 

23.1300670 

8.1180414 

.001869159 

536 

287.396 

153990656 

23.1516738 

8.1330963 

.001865672 

537 

288369  ' 

154a54153 

23.1732605 

8.1381447 

.001862197 

538 

'  3894-44 

155720873 

23.1948370 

8.  1331870 

.001858736 

539 

21)0531 

156590819 

23.2163735 

8.1383330 

.001855288 

540 

291600 

157464000 

23.2379001 

8.1433539 

.001851852 

541 

392681 

15&340421 

23.3594067 

8.1483765 

.001848429 

543 

293764 

159330088 

23.2808935 

8.1532939 

.001845018 

543 

294849 

10;  103007 

23.3023604 

8.1583051 

.001841631 

544 

295936 

160989184 

23.3238076 

8.1633102 

.001838335 

515 

297025 

161878635 

23.3452351 

8.1683092 

.001834862 

546 

298116 

163771336 

23.3666429 

8.1733020 

.001831502 

54?' 

299209 

163667323 

23.3880311 

8.1782888 

.001828154 

548 

300304 

104566592 

23.4093998 

8.1&33695 

.001824818 

549 

301401 

165469149 

23.4307490 

8.1882441 

.001821494 

550 

302500 

106375000 

23.4520788 

8.1932127 

.001818182 

551 

303601 

167384151 

23.4733892 

8.1981753 

.001814882 

553 

304704 

168196608 

23.4946802 

8.2031319 

.001811594 

553 

305S09 

169113377 

23.5159520 

8.2080825 

.001808318 

854 

306916 

170031464 

23.5372046 

8.2130271 

.001805054 

i   555 

308035 

170953875 

23.5584380 

8.2179657 

.001801802 

556 

309136 

171879616 

23.5796522 

8.2228985 

.001798561 

557 

310349 

173808693 

23.6008474 

8.22782.54 

.001795332 

558 

311364 

173741112 

23.6220236 

8.2337463 

.001792115 

273 


TABLE  VIII.— Continued. 


No. 

Squares. 

Cubes. 

Square 
Boots. 

Cube  Roots. 

Reciprocals. 

559 

312481 

174676879 

23.6431808 

8.2376514 

.001788909 

560 

313600    175616000 

23.6643191     8.2425706 

.001785714 

561 

314721    176558-i81 

23.6854386 

8.2474740 

.001782531 

562 

315844 

177504328 

23.7065392 

8.2523715 

.001779859 

563 

316969 

178453547 

23.7270210 

8.2572633 

.01)177  0199 

564 

318096 

179406144 

23.7486842 

8.2621492 

.001773050 

565 

319225  !  180362125 

23.7697286 

8.2670294 

.001769912 

566 

320350    181321496 

23.7907545 

8.2719039 

.001766784 

567 

321489 

182284263 

88.8117818 

8.2767726 

.001763668 

568 

322624 

183250432 

28.8327506 

8.2816355 

.001760563 

569 

323701 

184220000 

H&.S687309 

8.X&64928 

.001757469 

570 

324900 

185193000 

23.8746728 

8.2913444 

.001754386 

571 

326041 

180169411 

23.8956063 

8.2961903 

.001751313 

572 

327184 

187149248 

23.9165215 

8.  £010304 

.001748252 

573 

328329 

188132517 

23.9374184 

8.3058651 

.001745201 

574 

329476 

189119224 

2x3.9582971 

8.3106941 

.001742160 

575 

330625 

190109375 

23.97'91576 

8.3155175 

.001739130 

576 

331776 

191102976 

24.0000000 

8.3203353 

.001736111 

577 

332929 

192100033 

24.0208243 

8.3251475 

.001733102 

578 

334084 

193100552 

24.0416306 

8.32995-12 

.001730104 

579 

335241 

194104539 

24.0024188 

8.3347553 

.601727116 

580 

336400 

195112000 

24.Ca31891 

8.3395509 

.001724138 

581 

337561 

196122941 

24.1039416 

8.3443410 

.0017-21170 

582 

338724 

197137368 

24.1246762 

8.3491256 

.001718213 

583 

339889 

198155287 

24.1453929 

8.0539047 

.001715266 

584 

341056 

199176704 

24.1660919 

8.3586784 

.001712329 

585 

342225 

200201625 

24.1867732 

8.:  £634466 

.001709402 

586 

343:396 

201230056 

24.2074369 

8.8682065 

00170(1485 

587 

344569 

202262003 

24.2280829 

8.o729Gt>8 

.001703578 

588 

345744 

203297472 

24.2487113 

8.8777188 

.001700680 

589 

346921 

204336469 

24.2693222 

8.8824653 

.OU1G97793 

590 

348100 

205379000 

24.2899156 

8.3872065 

.001694915 

591 

349281 

206425071 

24.3104916 

8.3919423 

.001692047 

592 

350464 

207474688 

24.3310501 

8.  £966729 

.001089189 

593 

351649 

208527857 

24.3515913 

8.4013981 

.001686341 

594 

£52836 

209584584 

24.3721152 

8.4061180 

.001683502 

595 

354025 

210644875 

24.3926218 

8.4108326 

.001680672 

596 

355216 

211708736 

24.4131112 

8.4155419 

.001677852 

597 

356409 

2127761  73 

24.4335834 

8.4202460 

.001675048 

598 

357604 

213847192 

24.4540385 

8.4249448 

.001672241 

599 

358801 

214921799 

24.47'44765 

8.4296383 

.001669449 

600 

360000 

216000000 

24.4948974 

8.4348267 

.001666G67 

601 

361201 

217081801 

24.5153013 

8.4390098 

.001668884 

602 

362404    218167208 

24.5356883 

8.4436877 

.0016611.30 

603 

363609  ;  219256227 

24.5560583 

8.4483605 

.001058375 

604 

364816    220:348864 

24.5764115 

8.4530281 

.001C55629 

605 

366025 

221445125 

24.5967'478 

8.4576906 

.001652893 

606 

367236 

22254501(3 

24.  6170673 

8.4623479 

.001650165 

607 

368449 

223648543 

24.6373,00 

8.4670001 

.001(147446 

608 

369664 

224755712 

24.6576560 

8.4716471 

.001(544737 

609 

370881 

225866529 

24.6779254 

8.47'02892 

.001042036 

610 

372100 

226981000 

24.6981781 

8.-  1809261 

.001639344 

611 

373321 

228099131 

24.7184112 

8.485557$) 

.0(11036661 

612 

374544 

229220928 

34.7886338 

8.4901848 

.001633987 

613 

375769 

230346397 

24.7588368 

8.4948065 

.001631321 

614 

376996 

831475544 

24.7790234 

8.4994233 

.001028664 

615 

378225 

232608375 

24.7991935 

8.5040350 

.001626016 

616 

379456 

233744896 

24.8193473 

8.5086417 

.001(123377 

617 

380689 

234885113  i   24.83048-17 

8.5132435 

.001620746 

618 

381924 

236029032     24.KV.MH»5rt  !   8.517K403      .001618123 

619 

383161 

237J76659     24.H797IO'!     8..V,>24321      .001615509 

620 

384400 

238328000     24.8997992     8.5270189   i   .  001  (51  2903 

374 


TABLE  VIII.— Continued. 


No. 

Squares 

Cubes. 

Square 
Roots. 

Cube  Roots. 

Reciprocals. 

621 

385641 

239483061 

24.9198716 

8.5316009 

.001610306 

622 

386884 

240641848 

24.9399278 

8.5361780 

.001607717 

653 

388129 

241804367 

24.9599679 

8.  540750  1 

.001605136 

624 

389376 

242SJ70624 

24.9799920 

8.5453173 

.001602564 

625 

390625 

244140625 

25.0000000 

8.5498797 

.001600000 

626 

i  391876 

245314376 

25.0199920 

8.-J544372 

.001597444 

627 

!  393129 

246491883 

25.0399681 

8.5589899 

.001594896 

628 

394384 

247673152 

SB.  0599283 

8.5635377 

.001592357 

629 

395641 

248858189 

25.079S7'24 

8.5680807 

.001589825 

630 

396900 

250047000 

25.0998008 

8.5726189 

.001587302 

631 

398161 

251239591 

25.1197134 

8.5771523 

.001584786 

632 

399424 

252435968 

25.1396102 

8.5816809 

.001582278 

633 

4U0689 

253636137 

25.1594913 

8.5862047 

.001579779 

634 

401956 

254840104 

25.1793566 

8.5907238 

.001577287 

635 

403225 

256047875 

25.1992063 

8.5952380 

.001574803 

636 

404496 

257259456 

25.2190404 

8.5997476 

.001572327 

637 

405769 

258474853 

25.2388589 

8.6042525 

.001569859 

638 

407044 

259694072 

25.2586619 

8.6087'526 

.001567398 

639 

!  408321 

260917119 

25.2784493 

8.6132480 

.001564945 

640 

409600 

262144000 

25.2982213 

8.6177388 

.001562500 

641 

410881 

263374721 

25.3179778 

8.6222248 

.001560002 

642 

412164 

264609288 

25.3377189 

8.6267063 

.001557632 

643 

413449 

265847707 

25.3574447 

8.6311830 

.001555210 

644 

414736 

267089984 

25.3771551 

8.6356551 

.001552795 

645 

416025 

268336125 

25.3968502 

8.6401226 

.001550388 

646 

417316 

269586136 

25.4165301 

8.6445855 

.001547988 

647 

418609 

270840023 

25.4361947 

8.6490437 

.001545595 

648 

419904 

272097792 

25.4558441 

8.6534974 

.001543210 

649 

421201 

273359449 

25.4754784 

8.6579465 

.001540832 

650 

422500 

274625000 

25.4950976 

8.6623911 

.001538462 

651 

423801 

275894451 

25.5147016 

8.6668310 

.001536098 

652 

425104 

277167808 

25.5342907 

8.6712665 

.001533742 

653 

426409 

278445077 

25.5538647 

8.6756974 

.001531394 

654 

427716 

279726264 

25.5734237 

8.6801237 

.001529052 

655 

429025 

281011375 

25.5929678 

8.6845456 

.001526718  • 

656 

430*36 

282800416 

25.6124969 

8.6889630 

.001521390 

657 

431649 

283593393 

25.6320112 

8.6933759 

.001522070 

658 

432964 

284890312 

25.6515107 

8.6977843 

.001519757 

659 

434281 

286191179 

25.6709963 

8.7021882 

.001517451 

660 

435600 

287496000 

25.6904652 

8.7065877 

.001515152 

661 

436921 

288804781 

25.7099203 

8.7109827 

.001512859 

662 

438244 

290117528 

25.7293607 

8.7153734 

.001510574 

663 

439569 

291434247 

25.7487864 

8.7197596 

.001508296 

664 

440896 

292754944 

25.7681975 

8.7241414 

.001506024 

665 

442225 

294079625 

25.7875939 

8.7285187 

.0015037'59 

666 

443556 

295408296 

25.8069758 

8.7328918 

.001501502 

667 

444889 

2967'40963 

25.8263431 

8.7372604 

.001499250 

668 

446224 

298077632 

25.8456960 

8.7416246 

.001497'006 

669 

447561 

299418309 

25.8650343 

8".  7459846 

.001494708 

670 

448900 

300763000 

25.8843582 

8.750:3401 

.001492537 

671 

450241 

302111711 

25.  9036677 

8.7546913 

.001490313 

672 

451584 

303464448 

25.9229628 

8.75903*3 

.001488095 

673 

452929 

304821217 

25.9422435 

8.76,33809 

.001485884 

674 

454276 

306182024 

25.9615100  j 

8.7677192 

.00148:3680 

675 

455625 

307546875 

25.9807621 

8.7720532 

.001481481 

676 

450976 

308915776 

26.0000000 

8.7763830 

.001479290 

677 

458329 

310288733 

26.0192237 

8.7807084 

.001477105 

678 

459684 

311665752 

26.0384831 

8.7850296 

.001474926 

679 

461041 

313046839 

£6.0576284 

8.7893466 

.004472754 

680 

462400 

314432000 

26.0768096 

8.7936593 

.001470588 

681 

463761 

315821241 

26.0959767 

8.7979679 

.001468429 

682 

465124 

317214568 

26.1151297 

8.8022721 

.001466276 

375 


TABLE  VIII.-Confm««f. 


No. 

Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots. 

Reciprocals. 

683 

466489 

318611987 

26.1342667 

8.8065722 

.001464129 

681 

467-856 

320013504     26.1533937 

8.8108081 

.001401988 

685 

469225 

321419125 

26.1725047 

8.8151598 

.001459854 

686 

470596 

322828856 

26.1916017 

8.8194474 

.001457726 

687 

471969 

324242703 

20.2100848 

8.8237307 

.001455004 

088 

47-3;344 

325600672     26.2297541 

8.8280099 

.001453488 

689 

474721 

327082769 

26.2488095 

8.8322850 

.001451379 

690 

476100 

328509000 

26.2078511 

8.8365559 

.001449275 

691 

477481 

329939371 

20.2808789 

8.8408227 

.001447178 

692 

478864 

331373888 

20.3058929 

8.8450854 

.001445087 

693 

480249 

332812557 

26.3248932 

8:849:3440 

.001443001 

694 

481636 

334255384 

26.3438797 

8.8535985 

.001440922 

695 

483025 

£35702375 

26.3628527 

8.8578489 

.001438849 

696 

484416 

337153536 

26.3818119 

8.8630953 

.001430782 

697 

485809 

33860887-3 

26.4007576 

8.8003375 

.0014:347-20 

698 

487204 

340068392 

26.4190890 

8.87-05757 

.001432005 

699 

488601 

341532099 

26.4386081 

8.8748099 

.001430015 

700 

490000 

343000000 

26.4575131 

8.8790400 

.001428571 

701 

491401 

344472101 

26.4764046 

8.8832001 

.001420534 

702 

492804 

345948408 

20.4952820 

8.8874883 

.001424501 

703 

494209 

347428927 

20.5141472 

8.8917063 

.001422475 

704 

495616 

348913604 

20.5329983 

8.8959204 

.001420455 

705 

497025 

350402625 

20.5518361 

8.9001304 

.001418440 

708 

498436 

351895816 

26.5706605 

8.9043300 

.00141(5431 

707 

499849 

353393243 

26.5894716 

8.9085387 

.001414427 

708 

501264 

&54894912 

20.0082094 

8.9127369 

.001412429 

709 

502681 

350400829 

20  6270539 

8.9169311 

.001410437 

710 

504100 

38791^000 

26.6458252 

8.9211214 

.001408451 

711 

505521 

359425431 

26.6045833 

8.9358078 

.001400470 

712 

506944 

360944128 

20.0833281 

8.9294902 

.001404494 

713 

508369 

362467097 

20.7020598 

8.9330087 

.001402525 

714 

509796 

303994344 

20.7207784 

8.9378433 

.001400500 

715 

511225 

305525875 

2(5.7394839 

8.9420140 

.001398601 

716 

512656 

367061696 

20.7581703 

8.9401809 

.001396648 

717 

514089 

368601813 

20.77-08557 

8.9503438 

001394700 

718 

515524 

370146232 

20.7955220 

8.9545029 

.001:392758 

719 

516961 

371694959 

26.8141754 

8.9580581 

.001390821 

720 

518400 

373248000 

26.8328157 

8.9628095 

.001388889 

721 

519841 

374805361 

20.8514432 

8.9669570 

.(;01380963 

722 

521284 

37'6367048 

26.8700577 

8.9711007 

001385043 

723 

5227'29 

377933067 

20.8880593 

8.9752406 

.001:383120 

724 

524176 

379503124 

20.9073481 

8.97937-60. 

.OOK381215 

725 

525625 

381078125 

86.9258840 

8.9835089 

.001379:310 

726 

527076 

382657176 

26.944:3872 

8.9870373 

.001377410 

727 

528529 

384240583 

26.9629375 

8.9917020 

.  0013755  10 

728 

529984 

385828352 

26.9814751 

8.9958889 

0013730:$ 

729 

531441 

387420489 

27.0000000 

9.0000000 

.001371742 

730 

532900 

389017000 

27.0185122 

9.0041134 

.001369863 

731 

534:361 

890817891 

27.0370117 

9.0082229 

.001307'!W9 

732 

535824 

392223168 

27.0554985 

9.0123288 

.001306120 

733 

537289 

393832837 

27.0739727 

9.010430!) 

.001304256 

734 

538756 

395440904 

27.0924344 

9.0205293 

.001362398 

735 

540225 

897065375 

27.1108834 

9.0246339 

.001300544 

736 

541696 

398688258 

27.1293199 

9.0287149 

.001358(590 

737 

543169 

400816663 

27.1477439 

9.0328021 

.001356853 

738 

£44644 

40194727'2 

27.10<>ir>54 

9.0368857 

.001:355014 

739 

546121 

403583419 

27.1845514 

9.0409655 

.001353180 

740 

547600 

405224000 

27.2029410 

9.0450419 

.001  351  351 

741 

549081 

4068(59021 

27.221.-J1.V3 

!).(  1491  142 

.OOK349528 

742 

550564 

408518488 

87.2896769 

9.0531881 

.001:347709 

743 

552049 

410172407 

27.2580363 

9.0572482 

.001345895 

744 

553536 

411830784 

27.2703034 

9.0613098 

.001344086 

276 


TABLE  Vlll.— Continued. 


No. 

Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots. 

Reciprocals. 

745 

555025 

413493625 

27.2946881 

9.0653677 

.001342288 

746 

556516 

415160936 

27.3130006 

9.0694220 

.001340483 

747 

558009 

4168:32723 

27.3313007 

9.0734726 

.00133868S 

748 

559504 

418508992 

27.3495887 

9.0775197 

.001336898 

749 

561001 

420189749 

27.3678644 

9.0815631 

.001335113 

750 

562500 

421875000 

27.3861279 

9.0856030 

.001333333 

751 

564001 

423564751 

27.4043792 

9.0896392 

.Or]  331558 

752 

565504 

425259008 

27.4226184 

9.0936719 

001329787 

753 

567009 

426957777 

27.4408455 

9.0977010 

.001328021 

754 

568516 

428661064 

27.4590604 

9.1017265 

001326260 

755 

570025 

'430368875 

27.4772633 

9.1057485 

.001324503 

756 

571536 

432081216 

27.4954542 

9.1097669 

.001322751 

757 

573049 

433798093 

27'.  5136330 

9.1137818 

.001321004 

758 

574564 

435519512 

27.5317998 

9.1177931 

.001319261 

759 

576081 

437245479 

27.5499546 

9.1218010 

.001317523 

760 

577600 

438976000 

27.5680975 

9.1258053 

.001315789 

761 

579121 

440711081 

27.5862284 

9.1298061 

.001314060 

762 

580644 

442450728 

27.6043475 

9.1338034 

.001312336 

763 

582169 

444194947 

27.62-,4546 

9.1377971 

.001310616 

764 

583696 

445943744 

27.6405499 

9.1417874 

.001308901 

765 

585225 

447697125 

27.6586*34 

9.1457742 

.001307190 

7'66 

586756 

449455096 

27.6767050 

9.1497576 

.001305483 

767 

588289 

451217663 

27.6947648 

9.1587375 

.001303781 

768 

589824 

452984832 

27.7128129 

9.1577139 

.001302083 

769 

591361 

454756609 

27.7308492 

9.161686'J 

.001300390 

770 

592900 

456533000 

27.7488739 

9.1656565 

.001298701 

771 

594441 

458314011 

27.7068868 

9.1696225 

.001297017 

772 

595984 

460099648 

27.7848880 

9.1735852 

.001295337 

773 

597529 

461889917 

27.802S775 

9.1775445 

.001293661 

774 

599076 

463684824 

27.8208555 

9.1815003 

.001291990 

775 

600625 

465484375 

27.8388218 

9.1854527 

.001290323 

776 

C02176 

467288576 

27.8567766 

9.1894018 

.001288660 

777 

(503729 

469097433 

27.8747197 

9.1933474 

.001287001 

778 

605284 

470910952 

27.8926514 

9.1972897 

.001285347 

779 

606841 

472729139 

27.9105715 

9.2012286 

.001283697 

780 

608400 

474552000 

27.9284801 

9.2051641 

.001282051 

781 

609961 

476379.541 

27.9463772 

9.2090962 

.001280410 

782 

611524 

478211768 

27.9642U29 

9.2130250 

.001278772 

783 

613089 

480048687 

27.9821372 

9.2169505 

.001277139 

784 

614656 

481890304 

28.0000000 

9.2208726 

.001275510 

785 

616225 

483736625 

28.0178515 

9.2247914 

.001273885 

786 

617796 

485587656 

28.0356915 

9.2287068 

.001272265 

787 

619369 

48744:3403 

28.0535203 

9.2326189 

.001270648 

788 

620944 

489303872 

28.07ia377 

9.2365277 

.001269036 

789 

622521 

491169069 

28.0891438 

9.2404333 

.001267427 

790 

624100 

493039000 

28.1069386 

9.2443355 

.001265823 

791 

625681 

494913671 

28.1247222 

9.2482344 

.001264223 

792 

627264 

496793088 

28.1424946 

,9.2521300 

.001262626 

793 

628849 

498677257 

28.1602557 

9.2560224 

.001261034 

794 

630436 

500566184 

28.1780056 

9.2599114 

.001259446 

795 

632025 

502459875 

28.1957444 

9.2637973 

.001257862 

796 

633616 

504358336 

28.2134720 

9.2676798 

.001256281 

797 

635209 

506261573 

28.2311884 

9.2715592 

.001254705 

798 

636804 

508169592 

28.2488938 

9.2754352 

.001253133 

799 

638401 

510082399 

28.2665881 

9.2793081 

.001251564 

800 

640000 

512000000 

28.2842712 

9.2831777 

.001250000 

801 

641601 

513922401 

28.3019434 

9.2870440 

.001248439 

802 

643204 

515849608 

28.3196045 

9.2909072 

.001246883 

803 

644809 

517781627 

28.3372546 

9.2947671 

.001245330 

804 

646416 

519718464 

28.a548938 

9.2986239 

.001243781 

805 

648025 

521660125 

28.3725219 

9.3024775 

.001242236 

806 

649636 

523606616 

28.3901391 

9.3063278 

.001240695 

277 


TABLE  Vm.— Continued. 


No. 

Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots. 

Reciprocals. 

807 

651249 

525557943 

28.4077454 

9.3101750 

.001230157 

808 

652864 

527514112 

28.4253408 

!).  3140190 

.0012:37024 

809 

654481 

529475129 

28.4429253 

9.3178599 

.001230094 

810 

656100 

531441000 

28.4604989 

9.3216975 

.001  23451  M 

811 

657721 

533411731 

28.4780617 

9.3255320 

.00123:3040 

812 

659344 

535387328 

28.4956137 

9.3293034 

.001231527 

813 

660969 

537307797 

28.5131549 

9.3331910 

.001230012 

814 

662596 

539353144 

28.5300852 

9.3370167 

.001228501 

815 

664225 

541343375 

28.5482048 

9.3408386 

.001228094 

816 

665856 

543338496 

28.5657137 

9.3446575 

.0012-25190 

817 

667489 

545338513 

28.5832119 

9.3484731 

.00122J399U 

818 

669124 

547343432 

28.6000993 

J.  3522857 

.001222494 

819 

670761 

549353259 

28.6181760 

9.3560U52 

.001221001 

820 

672400 

551368000 

28.6356421 

9.3599016 

.001219512 

821 

674041 

553:387661 

28.6530976 

9.3037049 

.001218027 

822 

675684 

555412248 

28.6705424 

9.3675051 

.001210545 

823 

677329 

557441767 

28.  6871)?  tit) 

9.3713022 

.001215007 

824 

678976 

559476224 

28.7054002 

9.3750963 

.001213592 

825 

680625 

561515625 

28.7228132 

9.3788873 

.001212121 

826 

682276 

563559976 

28.7402157 

9.3820752 

.00121065  i 

827 

683929 

565609283 

28.7570077 

9.3804600 

.0012091M) 

828 

685584 

567003553 

28.7749891 

9.3902419 

.001207729 

829 

687241 

569722789 

28.7923601 

9.3940206 

.001200273 

830 

688900 

571787000 

28.8097206 

9.3977964 

.001204819 

831 

690561 

573836191 

28.8270706 

9.4015691 

.001203309 

832 

692224 

575930368 

28.8444102 

9.4053387 

.001201923 

833 

693889 

578009537 

28.8617394 

9.4091054 

.001200480 

834 

695556 

580093704 

28.8790582 

9.4128090 

.001199041 

835 

697225 

582182875 

28.8963666 

9.416G297 

.001  197005 

836 

698896 

584277056 

28.9130046 

9.4203873 

.001190172 

837 

700569 

586376253 

28,9809598 

9.4241420 

.001194743 

838 

702244 

588480472 

28.9482297 

9.4278936 

.001193317 

839 

703921 

590589719 

28.9654967 

9.4316423 

.001191895 

840 

705600 

592704000 

28.9827535 

9.4353880 

.00119047u 

841 

707281 

59482:3321 

29.0000000 

9.4391307 

.001189001 

842 

708964 

596947688 

29.0172363 

9.4428704 

.001187'64'3 

843 

710049 

599077107 

29.0344023 

9.4460072 

.001186240 

844 

712336 

601211584 

29.0516781 

9.4503410 

.001184831 

845 

714025 

60a351125 

29.0688837 

9.4540719 

.0011  8:3  132 

846 

715716 

605495736 

29.0860791 

9.4577999 

.001182033 

847 

717409 

607645423 

29.1032044 

9.4615249 

.0011  NOT.*  5 

848 

719104 

609800192 

29.1204396 

9.4652470 

.001179245 

849 

720801 

611960049 

29.1370046 

9.4689061 

.001177850 

850 

722,500 

614125000 

29.1547595 

9.4726824 

.001176471 

851 

724201 

616295051 

29.1719043 

0.4763957 

.0011750SS 

852 

725904 

618470208 

29.1890390 

9.4801061 

.001173709 

853 

727609 

620650477 

29.2061037 

9.4838136 

.001172333 

854 

729316 

622835864 

29.2232784 

9.4875182 

.00117'0900 

855 

731025 

625026375 

29.2408830 

9.4912200 

.001  1(59591 

856 

732736 

627222016 

29.2574777 

9.4949188 

.00110S-221 

857 

734449 

629422793 

29.2745623 

9.4980147 

.0011008(11 

858 

736164 

631628712 

29.2916370 

9.5023078 

.001165501 

859 

737881 

633839719 

29.3087018 

9.5059980 

.001104144 

860 

739600 

636056000 

29.3257566 

9.5096854 

.001162791 

861 

741321 

638277^381 

29.3428015 

9.5133699 

.001161440 

862 

743044 

640503928 

29.3598365 

9.5170515 

.001160093 

863 

744769 

642735647 

29.3768616 

9.5207303 

.001158749 

864 

746496 

644972544 

29.3938769 

9.5244063 

.001157407 

865 

748225 

647214625 

29.4108823 

9.5280794 

.001156069 

866 

749956 

649461896 

29.4278779 

9.5317407 

.001154734 

867 

751689 

651714363 

29.4448637 

9.5354172 

.001153403 

868 

753424 

653972032 

29.4618397 

9.5390818 

.001152074 

278 


TABLE  Vlll.— Continued. 


No. 

Squares. 

Cubes. 

Square 

Roots. 

Cube  Roots. 

Reciprocals. 

8G9 

755161 

656234909 

29.4788059 

9.5427437 

.001150748 

870 

756900 

658503000 

29.4957624 

9.5464027 

.001149425 

871 

758041 

000770311 

29.5127091 

9.5500589 

.001148106 

87'2 

700384 

003U54&48 

29.5290401 

9.5537123 

.001140789 

873 

762129 

00f>338017 

21).  5405734 

9.5573630 

.00114547'5 

874 

763876 

667027024 

29.50:34910 

9.5010108 

.001144165 

875 

765025 

00!)92J87'5 

29.5803989 

9.5646559 

.001142857 

876 

767376 

67'2221376 

29.ryj72i)7'2 

9.5082982 

.001141553 

877 

769129 

074c26133 

29.0141858 

9.5719377 

.001140251 

878 

770884 

670830152 

29.0310048 

9.57557'45 

.001138952 

879 

773641 

679151439 

29.0479342 

9.5792085 

.001137656 

880 

774400 

681472000 

29.6047939 

9.5828397 

.001136364 

881 

776161 

6837'97841 

29.6816442 

9.5804082 

.001135074 

883 

777924 

686128908 

29.6984848 

9.5900939 

.0011337'87 

883 

779689 

688405387 

29.7153159 

9.5937169 

.001132503 

884 

781456 

690807104 

29.7321375 

9.5973373 

.001131222 

885 

783225 

69315-1125 

29.7'489496 

9.0009548 

.001129944 

886 

7'84996 

695500456 

29.7057'521 

9.0045096 

.001128668 

887 

786769 

697804103 

29.7'825452 

9.6081817 

.001127396 

888 

7B8544 

700227072 

29.7993289 

9.6117911 

.001126126 

889 

790321 

702595309 

29.8101030 

9.6153977 

.001124859 

890 

792100 

704969000 

29.8328678 

9.6190017 

.00112,3596 

891 

793881 

7W347971 

29.8496231 

9.6220030 

.001122334 

892 

795664 

7'097'32288 

29.8003090 

9.6202016 

.001121076 

893 

797449 

712121957 

M.  8831056 

9.6297975 

.001119821 

894 

799236 

714510984 

29.8998328 

9.0333907 

.001118568 

895 

801025 

710917375 

29.9165506 

9.6369812 

.001117318 

896 

802816 

719323136 

29.9332591 

9.6405690 

.001116071 

897 

804609 

7217'34273 

29.9499583 

9.6441542 

.001114827 

898 

800404 

724150792 

29.9000481 

9.6477367 

.001113586 

899 

808201 

726572699 

29.9833287 

9.6513100 

.001112347 

900 

810000 

729000000 

30.0000000 

-9.6548938 

.001111111 

901 

811801 

731432701 

30.0100620 

9.0584684 

.001.109878 

902 

813604 

733870808 

30.0333148 

9.6020403 

.001108&47 

903 

815409 

736314327 

30.0499584 

9.6650096 

.001107420 

904 

817216 

738763264 

30.0065928 

9.6691762 

.0011(6195 

905 

819025 

741217625 

30.0832179 

9.6727403 

.001104972 

906 

820836 

743677416 

30.0998339 

9.6763017 

.001103753 

907 

822649 

746142643 

30.1164407 

9.6798604 

.001102536 

908 

824464 

748613312 

30.1:330383 

9.6834166 

.001101322 

909 

826281 

751089429 

30.1496269 

9.6869701 

.001100110 

910 

828100 

753571000 

30.1002063 

9.6905211 

.001098901 

911 

829921 

756058031 

30.1827765 

9.6940694 

.001097695 

912 

831744 

758550528 

30.1993377 

9.6976151 

.00109&491 

913 

833569 

761048497 

30.2158899 

9.7011583 

.001095290 

914 

835396 

763551944 

30.2324329 

9.7040989 

.001094092 

915 

837225 

766060875 

30.2489069 

9.7082309 

.001092896 

916 

839056 

768575296 

30.2654919 

„  9.7117723 

.001091703 

917 

840889 

771095213 

30.2820079 

9.7153051 

.001090513 

918 

842724 

773620632 

30.2985148 

9.7188354 

.001089335 

919 

844561 

776151559 

30.3150128 

9.7223631 

.001088139 

920 

846400 

778C88000 

30.a315018 

9.7258883 

.001086957 

921 

848241 

781229961 

30.3479818 

9.7294109 

.001085776 

922 

850084 

.  783777448 

30.3644529 

9.7329309 

.001084599 

923 

851929 

786330467 

30.3809151 

9.7364484 

.001083423 

924 

853776 

788889024 

30.39736*3 

9.7399634 

.001082251 

925 

a55625 

791453125 

30.4138127 

9.7434758 

.001081081 

926 

857476 

794022776 

30.4302481 

9.7469857 

.001079914 

927 

859329 

796597983 

30.4466747 

9.7504930 

.001078749 

928 

861184 

799178752 

30.4630924 

9.7539979 

.001077586 

929 

863041 

801765089 

30.4795013 

9.7575002 

.001076426 

930 

864900 

804357000 

30.4959014 

9.7610001 

.001075269 

279 


TABLE  VIII.-  r,,,, /,„,/,.,/. 


No. 

Cubes. 

Square 
Boot* 

Cube  Roots. 

I  ;<•<•!  procals. 

931 

80605440] 

9    -B4MI 

.IMIlll',  Jll  1 

932 

•,i»t.:,.- 

.001072961 

Ml 

81211 

9.  714845 

.001071811 

Ml 

Ml, 

;  inr, 

'.i     ,  i'.i;  1:5 

MB 

0878 

'.1    , 

69519 

936 

876096 

1941171 

19466 

937 

80.6104557 

MB 

w.>-  .»,  r  V  .» 

;.,',,  j 

939 

n  J  ,  ,    . 

940 

883600 

::u  ';:.:»  ii'.M 

S8611 

.i*}]i.> 

941 

87621 

.001062699 

942 

:jo.«:: 

9.8028036 

.001061571 

941 

944 

81807 

B2711 

.001060445 

.00101 

'.i  l.-j 

85&025 

8438 

::u  .; 

9.8i; 

.001(1 

946 

894916 

571180 

,i,:/j] 

.OOK 

949 

896809 

80.778 

"I  I  Oil 

948 

Kr.l'.i. 

9.82; 

.00101 

949 

900601 

KM'  i 

.OOK  - 

990 

902500 

30.8220700 

9.880476? 

.Odli  .' 

951 

904401 

906804 

WBHI)  no* 

tf4972 

(i  -:,  ;:.  i:,:, 

MB 

908909 

149818 

964 

910116 

!i  M  i-j:,:;(; 

.001041 

961 

912085 

!i  ,-  , 

I  I'li'l^  l:.'d 

956 

918906 

80.9192497 

1  irj.M) 

957 

915849 

876* 

854166 

15617 

.001044982 

958 

917764 

!1  7919 

80.0515751 

OOK 

959 

919681 

881!)',  !<)?.» 

::d  ft 

9!S614818 

960 

921600 

8HJ^: 

80  96i 

141667 

961 

9C.2 

(i-:,  1  1  1 

S77128 

;;|   (i|i;|:j  |K 

Mill  |1 

.001039501 

981 

898056847 

8llo 

MI:;:, 

.001088422 

964 

11844 

81.0488494 

I)  s  ,  • 

966 

,^1-J.') 

81.0844491 

19461 

.(.'UP 

9M 

988156 

lid!  : 

81.08( 

.(-dp 

907 

904831068 

81.0966286 

})>•• 

ii-.'i; 

9M 

90-30; 

:;i   i 

9.89 

969 

81.]:.' 

9.8055801 

.0011 

970 

940900 

U1-J673000 

81.14 

(i  89! 

.00101 

971 

942841 

915498611 

::i   !• 

'.i  '.  • 

.OOK 

972 

944784 

918880048 

81.1769146 

•j  9057817 

.00102W07 

971 

946799 

98111 

::i   li 

Ii  W.MIVfj 

.001027749 

974 

994010494 

81  .£01 

'.I  'ji: 

.Cdll.Vi  '.'.II 

979 

950696 

59876 

II   '.:! 

.001026641 

9M 

92971  J  1  id 

::i  941 

9.911 

OOK 

917 

81.251 

I)  I-' 

gn 

956484 

985441859 

ooi< 

979 

958441 

<.'.',:','.i 

81.S8I 

ii  981 

I  lud 

980 

960400 

{»H1920nO 

81,8040517 

I)  93 

,0010 

981 

969861 

94407614] 

•.Hiir, 

.ooion 

982 

!M')!W;('|P'-S 

:;!  ::: 

.00101 

Ii  94 

.00101 

B8904 

•,',|.", 

!i  941 

(Ki!d|r,;.'(',d 

9M 

17097 

9.9497479 

;  ooioi 

9M 

979198 

:;i   10 

.001014199 

'(K7 

96150 

I)  'i.  • 

.001018171 

US* 

976144 

'.I   '.i.'  !lK.'JHII 

.001019146 

9M 

97H121 

51669 

81981 

.OOlOlllM 

990 

980100 

00900 

;,  ;,(,i,;i;-,  |(| 

.(H)ldididl 

10096 

ddidd'MiH^ 

992 

9B40M 

:i;i;  11*1488    ! 

::i    IIMMI815 

iooioi 

TABLE  IX.— LOGARITHM  OF  NUMBERS  FROM  0  TO  1000. 


No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

0 

0 

00000 

30103 

47712 

60206 

69897 

77815 

84510 

90309 

95424 

10 

00000 

00432 

00860 

01284 

01703 

02119 

02530 

02938 

03342 

03743 

11 

04139 

04532 

04922 

05307 

05690 

06070 

06446 

06819 

07188 

07555 

12 

07918 

08279 

08637 

08990 

09342 

09691 

10037 

10380 

10721 

11059 

13 

11394 

11727 

12057 

12385 

12710 

13033 

13354 

13672 

13988 

14301 

14 

14613 

14922 

15229 

15533 

15836 

16137 

16435 

16732 

17026 

17319 

15 

17609 

17898 

18184 

18469 

18752 

19033 

19312 

19590 

19866 

20140 

16 

20412 

20683 

20952 

21219 

21484 

21748 

22011 

22272 

22531 

22789 

17 

23045 

23300 

23553 

23805 

24055 

24304 

24551 

24797 

25042 

25285 

18 

25527 

25768 

26007 

26245 

26482 

26717 

26951 

27184 

27416 

27646 

19 

27875 

28103 

28330 

28556 

28780 

29003 

29226 

29447 

29667 

298S5 

20 

30103 

30320 

30535 

30749  30963 

31175 

.31386 

31597 

31806 

32015 

21 

32222 

32428 

32633 

32838  33041 

33244 

33445 

33646 

33846 

34044 

22 

34242 

34439 

34635 

34830 

35025 

35218 

35411 

35603 

35793 

35984 

23 

36173 

36361 

36549 

36736 

36922 

37107 

37291 

37475 

37658 

37840 

24 

38021 

38202 

38382 

38561 

38739 

38916 

39094 

39270 

39445 

39619 

25 

39794 

39967 

40140 

40312 

40483 

40654 

40824 

40993 

41162 

41330 

26 

41497 

41664 

41830 

41996 

42160 

42325 

42488 

42651 

42813 

42975 

27 

43136 

43297 

43457 

43616 

43775 

43933 

44091 

44248 

44404 

44560 

28 

44716 

44871 

45025 

45179 

45332 

45484 

45637 

45788 

45939 

46090 

29 

46240 

46389 

46538 

46687 

46835 

46982 

47129 

47276 

47422 

47567 

30 

47712 

47857 

48001 

48144 

48287 

48430 

48572 

48714 

48855 

48996 

31 

49136 

49276 

49415 

49554 

49693 

49831 

49969 

50106 

50243 

50379 

32 

50515 

50651 

50783 

50920 

51055 

51189 

513-22 

51455 

51587  51720 

33 

51851 

51983 

52114 

52244 

52375 

52504 

5-2634 

52763 

52892  53020 

34 

53148 

53275 

53403 

53529 

53656 

53782 

53908 

54033 

54158  54283 

35 

54407 

54531 

54654 

54777 

54900 

55022 

55145 

55267 

55388  55509 

36 

55630 

55751 

55871 

55991 

56110 

56229 

56348 

56467 

56585 

56703 

37 

56820 

56937 

57054 

57171 

57287 

57403 

57519 

57634 

577-19  57863 

38 

5797'8 

58093 

58206 

58320 

58433 

58546 

58659 

58771 

58883  58995 

39 

59106 

59218 

59328 

59439 

59550 

59660 

59770 

59879 

59989  60097 

40 

60206 

60314 

60423 

60531 

60638 

60745 

60853 

60959 

61066 

61172 

41 

61278 

61384 

61490 

61595 

61700 

61805 

61909 

62014 

62118 

62221 

42 

62325 

62428 

62531 

62634 

62737 

62839 

62941 

63043 

63144  63246 

43 

63347 

63448  63548 

63649 

63749 

63849 

63949 

64048 

64147  64246 

44 

64345 

64444  64542 

64640 

64738 

64836 

64933 

65031 

65128 

65225 

45 

65321 

65418  65514 

65609 

65706 

65801 

65896 

65992 

66087  66181 

46 

66276 

66370 

66464 

66558  66652 

66745 

66839 

6693'2 

67025  67117 

47 

67210 

67302 

67394 

67486  67578 

67669 

67761 

67852 

67943  68034 

48 

68124 

68215 

68305 

68395 

68485 

68574 

68664 

68753 

68842 

68931 

49 

69020 

69108 

69197  69285  69373 

69461 

69548 

69636 

69723  69810 

50 

69897 

69984 

70070  70157 

70243 

70329 

70415 

70501 

70586 

70672 

282 


TABLE  ix— continued.— LOGARITHM  OF  NUMBERS  FROM  o  TO  1000. 


No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

51 

70757 

70842 

70927 

71012 

71096 

71181 

71265 

71349 

71433 

71517 

52 

71600 

71684 

71767 

71850 

71933 

72016 

72099 

72181 

72263 

72346 

53 

72428 

72509 

72591 

72673 

72754 

72835 

72916 

72997 

73078 

73159 

54 

73239 

73320 

73399 

73480 

73560 

73(53!) 

73719 

73799 

73878 

73957 

55 

74036 

74115 

74194 

74273 

74351 

74429 

74507 

745S6 

74663 

74741 

56 

74819 

74896 

74974 

75051 

75128 

75205 

75282 

75358 

75435 

75511 

57 

75587 

75664 

75740 

75815 

75891 

75967 

76042 

76118 

76193 

76268 

58 

76343 

76418 

76492 

76567 

76641 

76716 

76790 

7G864 

76938  77012 

59 

77085 

77159 

77232 

77305 

77379 

77452 

77525 

77597 

77610 

77743 

60 

77815 

77887 

77960 

78032 

78104 

78176 

78247 

78319 

78390 

78462 

61 

78533 

78604 

78675 

78746 

78817 

78888 

78958 

79029 

79099 

79169 

62 

79239 

79309 

79379 

79449 

79518 

79588 

79657 

79727 

79796 

79865 

63 

79934 

80003 

80072 

80140 

80209 

80277 

80346 

80414 

80482 

80550 

64 

80618 

80686 

80754 

80821 

80889 

80956 

81023 

81090 

81158 

81224 

65 

81291 

81358 

81425 

81491 

81558 

81624 

81690 

81757 

81823 

81889 

66 

81954 

82020 

82086 

82151 

82217 

82282 

82347 

82413 

82478 

82543 

67 

82607 

82672 

82737 

82802 

82866 

82930 

82995 

83059 

83123 

83187 

68 

83251 

83315 

83378 

83442 

83506 

83569 

83632 

836% 

83759 

83822 

69 

83885 

83948 

84011 

84073 

84136 

84198 

842(51 

84323 

84386 

84448 

70 

84510 

84572 

84634 

84696 

84757 

84819 

84880 

84942 

85003 

85065 

71 

85126 

85187 

85248 

85309 

85370 

85431 

85491 

85552 

85612 

85673 

72 

85733 

85794 

85854 

85914 

85974 

86034 

86094 

86153 

86213 

86273 

73 

86332 

86392 

86451 

86510!  86570 

86629 

86688 

86747 

86806 

86864 

74 

86923 

86982 

87040 

87099 

87157 

87216 

87274 

87332 

87390 

87448 

75 

87506 

87564 

87622 

87680 

87737 

87795 

87852 

87910 

87967 

88024 

76 

88081 

88138 

88196 

88252 

88309 

88366 

88423 

88480 

88536 

88593 

77 

88649 

88705 

88762 

88818 

88874 

88030 

88986 

89042 

8909H 

89154 

78 
79 

89209 
89763 

89265 
89818 

89321 
89873 

89376 
89927 

89432 
89982 

89487 
90037 

89542 
90091 

89597 
90146 

89653  j  89708 
90200  90255 

80 

90309 

90363 

90417 

90472 

90526 

90580 

90634 

90687 

90741  90795 

81 

90848 

90902 

90956 

91009 

91062 

91116 

91169 

91222 

91275  91328 

82 

91381 

91434 

91487 

91540 

91593 

91645 

.91698 

91751 

91803.  91855 

83 

91908 

91960 

92012 

92065 

92117 

92169 

92221 

92273 

92324  92376 

84 

92428 

92480 

92531 

92583 

92034 

92686 

92737 

92789 

92840  02891 

85 

92942 

92993 

93044 

93095 

93146 

93197 

93247 

93298 

93349,  93399 

86 

93450 

935004 

93551 

93601 

93651 

93702 

93752 

93802 

93852!  93902 

87 

93952 

94002 

94052 

94101 

94151 

94201 

94250 

94300 

94349;  94398 

88 

94448 

94498 

94547 

94596 

94645 

94694 

94743 

94792 

94841)  94890 

89 

94939 

94988 

95036 

95085 

95134 

95182 

95231 

95279 

95328;  95376 

90 

95424 

95472 

95521 

95569 

95617 

95665 

95713 

95761 

95809 

95856 

91 

95904 

95952 

95999 

96047 

96095 

96142 

96190 

96237 

96284 

96332 

92 

96379 

96426 

96473 

965201  96567 

96614 

96661 

96708 

96755  96802 

93 

96848 

96895 

96942 

96988  97035 

97081 

97128 

97174 

97220,'  97267 

94 

97313 

97359 

97405 

97451  97497 

97543 

97589 

97635 

976811  97727 

95 

97772 

97818 

97864 

979091  97955 

98000 

98046 

98091 

98137  98182 

96 

98227 

98272 

9S318 

98363  98408 

98453 

98498 

98543 

98588 

98632 

97 

98677 

96722 

98767 

98811!  98856 

98900 

98945 

98989 

99034 

99078 

98 

99123 

99167 

99211 

99255  99300 

99344 

99388 

99432 

99476 

99520 

99 

99564 

99607 

99651 

99695 

99739 

99782 

99826 

99870 

99913 

99957 

283 


NOTE  TO  TABLES  OF  TRIGONOMETRIC 
FUNCTIONS. 

In  the  following  Tables  the  values  of  Sines,  Cosines,  Tangents, 
Cotangents,  Versines,  and  Exsecants  are  carried  only  to  5  places 
of  decimals ;  the  Table  of  Secants  and  Cosecants,  however,  is 
given  to  7  places  of  decimals,  and  from  it  more  accurate  deter- 
minations of  the  Sines,  etc.,  may  be  obtained,  if  for  any  special 
purpose  they  be  required.  For,  by  Sees.  231  and  232, 

1  1  sec  A 

sin  A  —  —  cos  A  = -. ;  tan  A  =  — 


-.  ,                     y^uo  .0.  — -.  ,  ICI.LI  -a.  —  —               j, 

cosec  A                             sec  A  cosec  A 

1  cosec  A 

vers  A  =  1 — ; :   exsec  A  —  sec  A  —  1 ;  cot  A  =  —   .-. 

sec  A  sec  A 

284 


TABLE  X.— SINES  AND  COSINES.  . 


0° 

1° 

2° 

30         40 

/ 

Sine  !  Cosin 

Sine 

Cosin 

Sine 

Josin 

Sine 

Cosin 

Sine  Cosin 

~o 

TOOOOO  One7 

.01745 

.99085 

.03490 

.99939 

.05234 

99863  .06976 

.99756 

60 

1 

.  0029  One.   .01774 

.99984 

.03519 

.99938 

.05203 

99861  .07005 

.99754 

59 

2 

.00058  One.  \  .01803 

.99984 

.03548 

.99937 

.05292 

99860 

.070134 

.99752!  58 

3  1.00087:  One.  !  .01832 

.99983 

.03577 

.99936 

.05321 

99858 

.07063 

.99750!  57 

4  i.  00116  One.   .01862 

.99983 

.03606 

.99935 

.05:350 

99857 

.07092 

.99748  50 

5  .00145  One.   .01891 

.99982 

.03635 

.99934 

.05379 

99855 

.07121 

.99746  55 

6  .00175  One.   .01920 

.99982 

.03664 

.99933 

.05408 

99854 

.07150 

.99744  54 

7  .00204  One.   .01949 

.99981  .03693 

.99932 

.05437 

.99852 

.07179 

.99742  53 

8  .00233  One. 

.01978 

.99980  1.03723 

.99931 

.05466 

.99851 

.07208 

.99740  52 

9  .00262  One. 

.02007 

.99980  .03752 

.99930 

.05495 

.99849 

.07237 

.99738!  51 

iO 

.00291  One.   .02036 

.99979 

;.  03781 

.99929 

.05524 

.99847 

.07266 

.99736 

50 

11 

.  00320  !.  99999  !  .02065 

.99979 

.03810 

.99927 

.05553 

.99846 

.07295 

.99734 

49 

12 

.00349  .99999  .  02094  i  .99978 

.03839 

.99920 

.05582 

.99844 

.07324 

.99731 

48 

13 

.00378  .99999:  1.02123  ;.  99977 

1  .03868 

.99925 

!  .05611 

.99842 

.07353 

.997'29 

47 

14 

.00407  .99999  .02152 

.99977 

!  .03897 

.99924 

.05640 

.99841 

.07382 

.99727 

46 

15 

.00436  .99999  !:.  021  81 

.99976 

.03926 

.99923 

.05069 

.99839 

.07411 

.99725 

45 

16 

.  00465  .  99999  .  0221  1  i  .  99976 

.03955 

.99922 

.05698 

.99838 

.07440 

.99723 

44 

17 

.  00495  .  99999  .  02240  .  99975 

.03984 

.99921 

.05727 

.99836 

.07469 

.997'21 

43 

18 

.00524  .99999  .02209  .99974 

!  .04013 

.99919 

.05756 

.99834 

.07498 

.99719 

42 

19 

.00553  .99998 

.02298  .99974 

.04042 

.99918 

.05785 

.99833 

.07527 

.99716 

41 

20 

.00582  .99998; 

.02327 

.99973 

.04071 

.99917 

.05814 

.99831 

.07556 

.99714 

40 

21 

.00611  .99998  1.02356 

.99972 

.04100 

.99916 

.05H44 

.99829 

.07585 

.99712 

39 

22 

.00640  .9<)'.I98  .02385 

.99972;  .04129 

.99915  .05873 

.99827 

.07614 

.99710 

38 

23 

.00669  .99998  i  .02414 

.99971 

1.04159 

.99913  .05902 

.99826 

.(  7643 

.99708 

37 

24 

.  00698  .  99998  i  .  02443  i  .  99970 

.04188 

.99912  .05931 

.99824 

.07672 

.99705 

36 

25 

.00727  .99997  .02472 

.99969 

;  .04217 

.99911  .05960 

.99822 

.07701 

.99703  35 

26 

.00756.99997  .02501 

.99969 

i  .04246 

.99910;  .05989 

.99821 

.07730 

.99701 

34 

27 

.00785  .99997 

.02530 

.99968 

!  .04275 

.99909!  .06018 

.99819 

.07759 

.99699 

33 

28 

.00814:.  99997 

.02560  [.99967 

.04304 

.99907  i  .06047 

.99817 

.07788 

.99696!  32 

29 

.00844  .99996 

.025891.99966 

.04333 

.  99906  :!  .06076 

.99815 

.07817 

.99094 

31 

30 

.00873:.  99996 

.02618 

.99966 

.04362 

.99905  |  .06105 

.99813  .07846 

.99692 

30 

31 

.00902  '.99996 

.02647 

.99965 

.04391 

.99904  .06134 

.99812  .07875 

.99689 

29 

32 

.00931:.  99996 

.02676 

.99964 

.04420 

.99902  .06163 

.99810  .07904 

.99087 

28 

33 

.00960  .99995 

.02705 

.99963 

.04449 

.99901  .06192 

.99808  .07933 

.99685 

27 

34 

.00989  .99995 

.02734 

.99963 

1.04478  .99900  .00:J21 

.99800 

.07962 

.  99683  j  26 

35 

.01018  .99995 

.02763 

.99962 

.045071.99898  .06250 

.99804 

.07991 

.99080  25 

36 

.01047  .99995; 

.02792 

.99961 

.0-1536!.  99897;  .00279 

.99803 

.08020 

.99678;  24 

37 

.01076  .99994 

.02821 

.99960 

.04565 

.99890 

.00308 

.99801 

.08049 

.99670  23 

38 

.01105  .99994 

.02850 

.99959 

.04594 

.99894 

.06387 

.99799 

.08078 

.99673  22 

39 

.01134  .99994  .02879 

.99959 

.04023 

.99893!:.  06306 

.99797 

.08107 

.99671 

21 

40 

.01164.99993  .02908 

.99958  .04653 

.99892  .06395 

.99795 

.08136 

.99668 

20 

41 

.01  193  '.99993 

.02938 

.99957  .04682  .99890  .06424 

.99793 

.08165 

.99666  19 

42 

.01222  .99993 

.02967 

.99956  1.04711 

.99889  .00453 

.99792 

.08194 

.99001  1  < 

43 

.01251  .99992 

,02996 

.99955 

.04740 

.99888 

.00482 

.99790 

.08223 

.99001  IT 

44 

.01280  .99992 

.03025 

.99954 

.04769 

.99886 

.06511 

.99788 

.08252 

.99059 

10 

45 

.01309  .99991 

.03054 

.99953 

.04796 

.99885 

.06540 

.99786 

.08281 

.99057 

15 

46 

.01338  .99991 

.03083 

.99952!  .04827 

.99883  .06569 

.99784 

.08310 

.99054 

14 

47 

.01367  .99991 

.03112 

.99952  '.  04856 

.99882  .06598 

.99782 

.08339 

.99052 

13 

48 

.01396  .99990 

.03141 

.  99951  li.  04885 

.  99881  !!.  06627 

.99780 

.08368 

.99019 

12 

49 

.01425  .99990 

.03170 

.99950!  .04914 

.99879  .00056 

.99778 

.08397 

.99647 

11 

50 

.01454  .99989 

.03199 

.99949  .04943  ,.99878 

.00685 

.99776 

.08426 

.99644 

10 

51 

.01  483  '.99989 

.03228 

.99948  ''.04972  .99876 

.06714 

.99774 

.08455 

.99642 

9 

52 

.01513  .99989 

.03257 

.99947!  .05001 

.99875  1.067431.99772 

.08484 

.99039 

8 

53 

.01542  .99988 

.03286 

.99946  .05030 

.99873 

.06773  .99770 

.08513 

.99037 

7 

54 

01571  .99988 

.03316 

.99945;!.  05059 

.99872 

.06802 

.99708 

.08542 

.99035 

6 

55 

.01600  .99987 

.03345 

.99944  1|.  05088 

.99870 

:  .06831 

.99706 

.08571 

.99032 

5 

56  .01629  .99987  .03374 

.99943  1.05117 

.998(59  .Of  5860  .99764 

.08600 

.99630 

4 

57  .01658  .99986  .03403 

.99942  .05146 

.99867  .06889.99762 

.08029 

.99627  3 

58  |  .01687  .999SO 

.03432 

.99911  .05175 

.  99866  '  .00918  .99700 

.08658 

.99625  2 

59 

.01716  .<t<)<IS.-> 

.03461 

.99940  .05205 

.99804  .00947  .99758 

.osr,S7  .99022  1  • 

60 

.01745  .99985 

.03490 

.99939  .0;V>;54 

.!)<!80:!  .()0!)70  .99756 

.08716!.  99619  |  0 

/ 

Cosin  i  Sine 

Cosin 

Sine  i  Cosin 

Sine  jjCosinj  Sine 

Cosin  Sine 

/ 

89° 

88°       87°      86° 

85° 

285 


TABLE  X. -SINES   AND   COSINES. 


5° 

6° 

7°        8e        9° 

Sine 

Cosin 

Sine  |  Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

t 

~0  1.08716  !  99619 
1  .08745  .99617 

.10453  799152 
.10482  .9941!) 

712187  .D'.rJ.Y, 
.12216  .99251 

:  13917 
.13946 

.99027  .15043 
.99023  .15(572 

798769 

.9876-4 

60 

59 

2  i  .08774 

.99614 

.10511 

.99446 

.12245 

.99248 

.13975 

.99019 

.15701 

.98760 

58 

3  i  .08803 

.99612 

.10540  .99443 

.12274 

.99244 

.14004 

.99015 

.15730 

.98755 

57 

4  .08831 

.99609 

.10569 

.99440  .12302 

.99240 

.14033 

.99011 

.  1575C 

98751 

56 

5  .08860 

.99607 

.10597 

.99437 

.  12331  |  .  99237  .  14061  i  .  99000 

.15787 

.9874(5 

55 

6  .08889 

.99604 

.10626 

.99434 

.12360.99233  .14090 

.99002 

.15816 

.98741 

54 

7  i  .08918 

99602 

.10655 

.99431 

.12389  .99230 

.14119 

.98998 

.  15845 

98737 

53 

8  ;  .08947 

.99599 

.10684 

.99428 

.12418^.99226 

.14148 

.98994 

.15873 

.98732 

52 

9  .08976  .99596 

.10713  .99424 

.12447;.  99222 

.14177 

.98990 

.15902 

.98728 

51 

10  .09005  .99594 

I 

.10742  .99421 

.124761.99219 

.  14205  j.  98986 

.15931 

.98723 

50 

11 

.090341.99591 

.10771 

.99418 

.12504  '.99215 

.14234 

.98982 

.159591.98718 

49 

12 

.090631.99588 

.10800  .99415 

.12533  .99211 

.M203 

.989781  .15988 

.98714 

48 

13  .09092 

.99586 

.10829  .99412 

.12562  .99208 

.14292 

.98973  .16017 

.98709 

47 

14 

.09121 

.99583 

.10858  .99409 

.12591 

.99204 

.143201.88969  .10040 

.98704 

46 

15 

.09150 

.99580 

.10887  .99406 

.12620  .99200 

.14349 

.UWM55  .10074 

.98700 

45 

16  .09179 

.99578 

.10916 

.99402 

.  12649  i.  99197 

.14378 

.98961 

.10103 

.98095 

44 

17  .09208 

.99575 

.10945 

.99399 

.12678 

.99193 

.14407 

.98957 

.16132  .98090 

43 

18  .09237 

.99572 

.10973  .9939(5 

.13706 

.99189 

.14436 

.98953  .16160 

.98080 

42 

19  .092(56 

.99570 

.11002  .99393 

.12735 

.99186 

14464 

.98948  .16189 

.98681 

41 

20  .09295 

.99567 

.11031  .99390 

.12764 

.99183. 

.14493 

.98944  .16218 

.98676 

40 

21  .09324 

.99564 

.11060 

.99386 

.12793 

.99178' 

.14522 

.98940:  .16246 

.98671 

39 

22  .09353 

.99562 

.11089 

.99383 

.  1  2822 

.99175 

.14551 

.98930  .16275 

.98667 

38 

23  .09382 

.99559 

.11118 

.99380 

!  12851 

.99171 

.14580 

.98931  .16304 

.98002 

37 

24  .09411 

.99556 

.11147 

.99377 

.12880 

.99167: 

.14008 

.98927  .16333 

.98057' 

36 

25  .09440 

.99553 

.11176 

.99374 

.12908 

.99163 

.14637 

.!I8S>23  .16361 

.JW552 

35 

26  .09469 

.99551 

.11205 

.99370 

.12937 

.99160 

.14666 

.9891  9  .16390 

.98648 

34 

27 

.09498 

.99548 

.11234 

.99367 

.12966 

.99156 

.14695 

.989141  .16419 

.98(543 

33 

28 

.09527 

.99545 

.11203 

.99364 

.12995 

.99152 

.14723 

.98910 

.16447 

.<(SO:!S 

32 

29 

.09556 

.99542 

.11291 

.13024 

.99148 

.14752  .98906 

.  16476 

31 

30 

.09585 

.99540 

.11320 

!  99357 

.13053 

.99144 

.14781 

.98902;  .16505 

.'98029 

30 

31 

.09614 

.99537 

.11349 

.99354 

.13081 

.99141 

.14810 

.98897 

.16533 

.98624 

29 

32 

.09642 

.99534 

.11378 

.99351 

.13110 

.99137' 

.14838 

.16502 

.98019 

28 

33 

.09671 

.99531 

.11407 

.99347 

.13139 

.99133 

.14867 

!  98889 

.10591 

.98014 

27 

34 

.09700 

.99528 

.11436 

.99344 

.18168 

.99129 

.14896 

.98884 

.1(56,20 

26 

35 

.09729 

.99526 

.11465 

.99341 

.13197 

.99125 

.14925 

.98880 

.16648 

!  98604 

25 

36 

.09758 

.99523 

.11494 

.99337 

.132261.99122 

.14954 

.98876 

.16677 

.98(500 

24 

37 

.09787 

.99520 

.11523 

.99334 

.132541.99118 

.14982 

.98871 

.16706 

.98595 

23 

38 

.09816 

.99517 

.11552 

.99331 

.13283 

.99114 

.15011 

.98867 

.16734 

.9S590 

22 

39 

.09845 

.99514 

1.11580 

.90327 

.13312 

.99110 

.15040 

.98863 

.16763 

.9S5S5 

21 

40 

.09874 

.99511 

1.11609 

.99324 

.13341 

.99106 

.15069 

.98858 

.16792 

.98580 

20 

41 

.09903 

.99508 

.11638 

.99320 

.13370 

.99102^ 

.15097 

.98854 

.16820 

.98575 

19 

42 

.09932 

.<a~>oo 

.11667 

.99317 

.133!)9 

.99098 

.15126 

.98849 

.1(5849 

.9857'0 

18 

43 

.09961 

.99503 

.11696 

.99314 

.13427 

.99094 

.15155 

.98845 

.16878 

.98565 

17 

44 

.09990 

.99500 

.11725 

.99310 

.13456 

.99091 

.151841.98841 

.1690(5 

.98561'  16 

45 

.10019 

.99497 

.11754 

•99307 

.13485 

.99087 

.15212 

.98836 

.16935 

.98556 

15 

46 

.10048 

.99494 

.11783 

.99:303 

.  13514 

.99083 

.15241 

.JISKJ2 

.16964 

.98551 

14 

47 

.10077 

.99491 

.11812 

.99:300 

.13543 

.99079 

.15270 

.1)8827 

.10992 

.98546 

13 

48 

.10106 

.99488 

.11840 

.99297 

.13572 

.99075; 

.15299 

.98823 

.17021 

.98541 

12 

49 

.10135 

.99485 

.11869 

.99293 

.13600 

.99071 

.15327 

.98818  .17050 

.98536 

11 

50 

.10164 

.99482 

1  .11898 

.99290 

.1:3029  .<)<)007 

.15356 

.98814 

;.  17078 

.98531 

10 

51 

.10192 

.99479 

'  .11927  .99286 

.13658  '.99063 

.15385 

.98809 

.17107 

.98526 

9 

52 

.10221 

.99476 

.11950  .99283 

.13687  .99059 

.15414 

.<)8S05  .17130 

.98521 

8 

53 

.10250 

.99473 

i  .11985  .9!)279 

.13716  .'0055 

.15442 

.98800  .17164 

.98516 

7 

54 

.10279 

.99470 

!.  12014 

.99276 

.13744 

.99051 

.15471 

.98796 

.17193 

.98511 

6 

55 

.10308 

.99467 

.12043 

.99272 

.13773 

.99047 

.15500 

.98791 

.17222 

.98506 

5 

56 

.10337 

.994(54 

.12071 

.99269 

.13802 

.99043 

.15529 

.98787 

.17250 

.98,501 

4 

57 

10366 

.99461 

.12100 

.1:3831  .99039 

.15557 

.98782 

!  .17279 

.98496 

3 

58 

.10395 

99458 

.121291.99262  .13860  .99035 

.15586 

.98778 

.17308 

.98491 

2 

59 

.104241.99455 

.12158  .99258  .13889  .99031  .15615 

.98773  .17336 

.98486 

1 

60 

.10153  .99452 

.12187  .99255 

.13917  .99027  .15643 

.98769  •.  17365: 

.98481 

J) 

"7 

Cosiu  j  Sine 

Cosin  Sine 

Cosiu  Sine 

Cosin 

Sine  Cosin 

Sine 

. 

84° 

83°       82» 

81° 

80° 

28G 


TABLE  X.— SINES   AND  COSINES. 


10°    ||    11°       12° 

13°    ||    14° 

Sine 

Cosin 

Sine  Cosin 

Sine  Cosin 

Sine  Cosin 

Sine 

Cosin 

0 

.17365 

.98481 

.19081 

.98163 

.20791  .!>7S15 

.22495  .97437 

.24192 

.97030 

60 

.17393 

.98476  .19109 

.98157 

.20820  .97809 

.22523  .974:30 

!  .24220 

.97023 

59 

2 

.17'422 

.98471  .19188 

.98152  .20848  .97803 

.  22552  ;.  97424 

.2424!) 

.97015 

58 

3 

.17'451 

.98466  i.  191  67 

.98146  1.20877  .97797 

.22580  .97417 

1  .24277 

.97008 

57 

4 

.17479 

.98461 

.19195 

.98140 

.20905  .97791 

.22608  .97411 

.24305 

.97001 

56 

5 

.17'508 

.98455 

.19224 

.98135 

.20'j:$3  .97784 

.22637  .97404 

.24333 

.96994 

55 

6 

.17'537 

.98450 

.1925.2 

.98129  .20962 

.9777'8 

.22665 

,07398 

.24362 

.96987 

54 

7 

.17565 

.98445 

.19281 

.98124 

.20990 

.97772 

.22693 

.  97391  ii  .24390 

.96980 

53 

8 

.17591 

.98440 

.19309 

.98118 

.21019 

.97766 

22722 

.97384!  .24418 

.96973 

52 

9 

.17623 

.98435 

.  19338 

.98112 

.21047 

.97760 

.22750  .97378!  .24446 

.96966 

51 

10 

.  17651 

,98480 

.19366 

.98107 

.21076 

.97754 

.2277«  .97371 

.24474 

.96959 

50 

11 

.17680 

.98425 

.19395 

.98101 

.21104 

.97748 

.22807  .97365 

.24503 

.96952 

49 

.17708 

.98420 

.19423 

.98096 

.21132 

.97742 

.22835 

.97:358 

.24531 

.96945 

48 

13 

.17737 

.98414 

.19452 

.98090 

.21161 

.97735 

.22863  .97351 

.24559 

.96937 

47 

14 

.17766 

.98109 

.19481 

.98084 

.21189 

.97729 

.22892  .97345 

.24587 

.96930 

46 

15 

.17794 

.  98404  .19509 

.98079 

.21218 

.97723 

.22920  .97338  .24615 

.96923  45 

16 

.17823 

.98399  .195J38 

.98073 

.21246 

.97717 

.22948  i.  97331  i!  .24644 

.96916 

44 

17 

.17852 

.98394! 

.19566 

.98067 

.21275 

.97711 

.22977 

.97323  .24672 

.96909 

43 

18 

.17880 

.98389  .19595 

.98061 

.21303 

.97705 

.23005 

.  97318  ii  .247'(K) 

.96902 

42 

19 

.17909  .98:38311.  19623 

.98056  .21331 

.97698 

.23033 

.97311  .24728 

.96894 

41 

20 

.17937 

.98378  |.19052 

.93050  .21360 

.97692 

.23062 

.  97304  |j  .24756 

.96887 

40 

21 

.17966 

.98373 

.19680 

.98044 

.21388 

.97686 

.23090 

.97298  .24784 

.96880 

39 

22 
23 

.17995 
.18023 

.98368 

.98362 

.19709 
.19737 

.98039 
.93033 

.21417 
.21445 

.97680 
.97673 

.23118 
.23146 

.972911  .24813 
.97284  .24841 

.96873  38 
.96866!  37 

24 

.18052 

.98357! 

.19766 

98027 

.21474 

.97667 

.23175 

.97278 

.248691.96858 

36 

25 

.  18081 

.98352: 

.19794 

93021 

.21502 

.97661 

.23203 

.97271 

.24897 

.96851 

-35 

26 

.18109 

.98347 

.19823 

98016  .21530 

.97655 

.232311.97264 

.24925 

.96844 

34 

27 

.18138 

.98341 

.19851 

98010  .21559 

.97648 

.23200 

.97257 

.24954 

.96837 

33 

28 

.18166 

.98336 

.19880 

98004  ;  .21587 

.97642 

.23288 

.97251  !  .24982 

.96829 

32 

29 

.  18195 

.98331 

.19908 

97993  !  .21616 

.97636  .23316 

.  97244 

.25010 

.96822 

31 

30 

.18224 

.98325 

.19937 

97992  .21644 

.97630 

.23345 

.97237 

.25038 

.96815J  3° 

31 

.18252 

.98320 

.19965 

97987'  .21672 

.97623 

.23373 

.97230 

.25066 

.96807 

29 

32 

.18281 

.98315 

.19994 

97981  .21701 

.97617 

.23401 

.97228 

.25094 

.96800 

28 

33 

.18309 

.98310  ;  .20022 

9797'5  .21729 

.97611 

.23429 

.97217 

.25122 

.96793 

27 

34 

.18338 

.93304  .20051 

97969  .21738 

.97604 

.23458 

.97210  !  .25151 

.96786 

26 

35 

.18367 

.98299 

.20079 

97963  .21786 

.97598 

.23486 

.97208  .25179 

.96778!  25 

36 

.  18395 

.93294  .23108 

97958  !  .21814 

.97592 

.23514 

.97196  .25207 

.96771 

24 

37 

.18424 

.98288  i  .20136 

97952  .21843 

.97585 

.23542 

.97189  .25235 

.  96764  i  23 

38 

.18452 

.98283  .20165 

97946  !  .21871 

.97579  .23571 

.97182  i  .25263 

.96756 

22 

39 

.18481 

.98277  .20193 

97940  .21899 

.!)7573  .23599 

.97176:  .25291  '.96749 

21 

40 

.18509 

.98272  .20222 

97934  .21928 

.97566 

.23627 

.9716U  .  25320  j.  96742 

20 

41 

.18538 

.98267 

.20250 

97923  .21956 

.97560 

.23656 

.97162  .2,5348  '.96734  19 

42 

.18567 

.98261  .20279;.  97922  i  .21985 

.97553 

.23684 

.97155  .25376  .96727 

18 

43 

.18595 

.98256  .20307 

.97916  .22013  .97547 

.23712 

.97148  .254041.96719 

17 

44 

.18624 

.98250  .20336 

.97910  .22041!.  97541  ' 

.23740 

.97141  .254321.96712 

16 

45 

.18652 

.98245  .20364 

.97905  .22070  .UIWH 

.23769 

.97134  .  25460  J.9G705 

15 

46 

.18681 

.98240,  .20393 

.97899  .22093:.  97528 

.23797 

.97127  .254881.96697 

14 

47 

.  18710 

.98234  .20421 

.97893  .22126  .97521 

.23825 

.97120  .255161.96690 

13 

48 

.18738 

.93-2!)  .20450  .97887  .  22155  1  .  97515 

.23853 

.97113:  .  25545  i.  96682 

12 

49 

.  18767 

.'13223  .2047.3  .97881  :  .22183  .97508 

.2:3882 

.97106"  .25573  .96675 

11 

50 

.18795 

.98218  i  .20507  1  .97875  j  i  .22212  .97502  .23810 

.97100  .25601  .96667 

10 

51 

.18824 

.  98212  .  20535  .  97869  .  22240  .  97496  .  23938 

.97093!  .25629  .96660 

9 

52 

.18852 

.98207  .20563  .97863  .22268  .97489  .2396") 

.97086  .25657  .96653 

8 

53 

.18881 

.98201  .20592  .97857  .22297  .97483  .23995 

.97079  .25685  .96645 

7 

54 

.18910 

.98196  .20620  .97851  .22325  .97476  .24023 

.97072  .25713  .96638 

6 

55 

.18938 

.98190  .20649  .97845  .22353  .97470  .24051 

.97065  .25741  .966:30 

5 

56 

.  18967 

.98185!  -.20677  .97839  .22382  .97463  .24079.97058  .25769.96623 

4 

57 

.  18995 

.!»S17!)  .20706  .97833  .22410  .97457  .24108  .97051  .25798  .96615 

3 

58 

.19024 

.98174'  .207:34  .97827  .22438  .97450  .24136  .97044  .25826  .96608 

2 

59 

.19052 

.  '.  M  i  (  iS  .  20763  .  97821  '•  .  22467  .  97444  .  24  1  64  .  97037  .  25854  .  96600 

1 

60 

.19081 

.98163J  .20791  .97815  .22495  .97437  .24192  .  97030  ;  .  25882  .96593 

0 

j  Cosin 

Sine  j  Cosin  i  Sine  Cosin  Sine 

Cosin 

Sine  Cosin  Sine 

J 

79° 

78°       77°       76°       75° 

387 


TABLE  X.— SINES   AND  COSINES. 


15° 

16°       17° 

!    18° 

19° 

Sine  Cosin 

Sine 

Cosin  Sine  j  Cosin 

Sine  Cosin 

Sine 

Cosin 

0 

.25882  .  96593  'j.  27564 

.96126  .29237 

.95630 

.30902 

.95106 

.82567 

.0155->  60 

1 

.25910  .96585  ;  .27592 

.96118  .21W!i.j 

.95622 

.30929 

.95CJ7 

.82584  .945421  59 

2  .35938  .96578  .27620 

.96110  .29293 

.95613 

.30957 

.95088 

.32612 

.915:5:!  58 

8  .95966  .9(5570  .27(548 

.96102  .29321 

.95605 

.30985 

.95079 

.32639 

.9452:3  57 

4  !.  25994  .96562  .27676 

.96094  .29348 

.95596 

.31012 

.95070 

.32667 

.94514  56 

5  .26022  .96555  .27704 

.96086  .29376 

.95588 

.31040 

.95061 

.32694 

.94504  55 

6  .26050  .96547  .27731 

.96078  .29404 

.95579 

.31068 

.95052 

.32722 

.94495  54 

7  1.  26079  !.  96540  I  .27759 

•96070  .29432 

.95571 

.31095 

.95043 

.32749 

.94485  53 

8 

.26107  .96532  |  .27787 

.96062  .294601.95562 

.31123 

.95033 

.32777 

.94476 

52 

9 

.26135  .96524  !  .27815 

.96054  .29487 

.9555  i 

.31151 

.95024 

.32804 

.94466  51 

10 

.26163:.  96517  .27843 

.96046  .29515  .95545 

.31178 

.95015 

.32832 

.94457 

50 

11 

.26191  .96509  .27871 

.96037  .29543 

.95536 

.31206  .95006 

.32859 

.94447 

49 

12 

.26219:.  96502  .27899 

.96029  .29571 

.95528 

.81238  .94997 

!  32887 

.944:38 

48 

13 

.26247  .96494  !  .27927 

.96021  .29599 

.95519 

.31261  .94988 

.32914 

.94428 

47 

14 

.26275  .96486  .27955 

.96013  .29626 

.95511 

.31289  .94979 

.32942 

.94418 

46 

15 

.26303  .96479  .27983 

.96005  .29654 

.95502 

.31316  .94970 

.32969 

.94409 

45 

16 

.26331  .96471  .28011 

.  95997  :  .29682 

.95493 

.31344  .94961 

.32997 

.94399 

44 

17 

.26359  .96463 

.28039 

.95989  .29710 

.95485 

.31372  .91953 

.33024 

.94390 

43 

18 

.2(3387  .96456 

.28067 

.95981  .29737 

.9.5476 

.81899  .94948 

.33051 

.948801  42 

19 

.26415  .96448 

.28095 

.95972  .29765 

.95467 

.31427  .94933 

:  .33079 

.94370  41 

20  .26443  .96440 

.28123 

.95964  .29793 

.95459 

.31454 

.94924 

.33106 

.94361  40 

21 

.26471 

.96433 

.28150 

.951)56  .29821 

.95450 

.31482 

.91915 

.33134 

.94351 

39 

22 

.26500 

.96425 

.28178 

.95948:  .29849 

.95441 

.31510 

.94906 

.33161 

.94.'3W 

38 

23 

.26528 

.96417 

.28206 

.95940  .29876 

.95433 

.31537 

.94897 

.33189 

.94332 

37 

24 

.26556  .96410 

.28234 

.95931  .299041.95424 

.31565 

.94888 

.33216 

.94822 

86 

25 

.26584  .96402 

.28262 

.95923 

.29932 

.95415 

.31593 

.94878 

.33244 

.94313;  35 

26 

.26612  .96394 

.28290 

.959151  .29960 

.O.Vldi' 

.31620 

.94869 

.33271 

.9  1303  34 

27 

.26640 

.96:386 

.28318 

.95907 

.29987!.  95398 

.31648 

.94860 

.33298 

.91-20:]  3:3 

28 

.26668  .96379 

.28346 

.95898 

.30015 

.95389 

.31675 

.94851 

.33326 

.912S1  33 

29 

.26696!.  96371 

.28374 

.95890 

.30043 

.95:380 

.31703 

.94842 

.33353 

.91271  31 

30 

.26724  .96363, 

.28402 

.95882 

.30071 

.95372 

|  .31730 

.94832 

.88881 

.94264 

30 

31 

.26752 

.96355 

.28429 

.95874 

.30098 

.95363 

1  .31758 

.94823 

.33408 

.94251 

29 

32 

.26780 

.96347 

.28457 

.95865 

i  .30120 

.95351 

.31786 

.94814 

.33436 

.942451  28 

33 

.26808 

.96340 

.28485 

.95857 

.30154 

.95345 

.31813 

.94805 

•33463 

.94235 

27 

34 

.26836 

.96:332 

.28513 

.95849 

.30182 

.95337 

.31841 

.94795 

.33490 

.94255 

26 

35 

.26864 

.96324 

.285411.95841 

.30209 

.95328 

i  .31868 

.94786 

.33518 

.94215 

25 

36 

.26892 

.96316 

.28569 

.95832 

.30237 

.95319 

i  .31896 

.94777 

.33545 

.94206 

24 

37 

.26920 

.96308 

.28597 

.95824 

.30265 

.95310 

i  .31923 

.94768 

.33573 

.91196  J.':} 

38 
39 

.26948  .96301 
.26976  .90293 

.28625 
.28652 

.95816 
.95807 

.30292 

.30320 

.95301 

.95293 

|  .31951 

1  .31979 

.94758 
.94749 

.  33600  j.  941  86  j  23 
.33627  .94176]  21 

40 

.27004 

.96285 

.28680 

.95799 

j  .30348 

.95284 

!.320QG 

.94740 

.3:3655 

.94167  20 

41 

.27032 

.96277 

.28708 

.95791 

.30376 

.95275 

.32034 

.94730 

.83682 

.94157  19 

42 

.27060 

.96269 

.28?'36 

.95782 

.30403 

.952(5(5 

'  .32001 

.94721 

.88710 

.94147  H 

43 

.27088 

.96261 

.28764 

.95774 

.30431 

.95257 

•  .32089 

.94712 

.38737 

.94137  17 

44 

.27116 

.96253 

.28792 

.95766 

.30459 

.95248 

:  .32116 

.94702 

.33764 

.941271  16 

45 

.27144 

.96246 

.28820 

.95757!  .30486 

.95240 

.32144 

.94693 

.33792 

.9111*  15 

46 

.27172 

.96238 

.28847 

.95749!  .30514 

.95231 

!  .32171 

.94684 

.33819 

.94108 

14 

47 

.27200 

.96230 

.28875 

.95740 

.30542 

.95222 

.32199 

.94674 

.33846 

.91098 

13 

48 

.27228 

.96222 

.28903 

.95732 

.30570 

.95213 

.32227 

.94665 

.83874 

.9  tOSS 

12 

49 

.27256 

.96214; 

.28931 

.95724 

.30597 

.95201 

.82254 

.91656 

33901 

.94078 

11 

50 

.27284 

.96206 

.28959 

.95715 

.30625 

.95195 

.32282 

.9464o' 

.33929 

.94068 

10 

51 

.27312 

.96198 

.28987 

.95707 

.306531.95186 

i  .32309 

.94637 

.33956 

.940.58 

9 

52 

.27340 

.96190 

.29015 

.95698 

.30680 

.95177 

.32337 

.94627 

.3:398:3 

.91049 

8 

53 

.27368  .96182 

.29012 

.95690 

.30708 

.95168 

.32364 

.94618  .34011 

.94039 

7 

54 

.27396  .96174 

.29070 

.95681 

.30736 

.95159 

.32392 

.94609  .34038 

.94029 

6 

55 

.27424  .96166 

.29098 

.95673 

.30763 

.95150 

.32419 

.91599  .34065 

.94019 

5 

56 

.27452  .{Mil  58 

.29126 

.95661  .30791 

.95142 

.82447 

.94590  .34093 

.94009 

4  ! 

57 

.27480  .96150 

.39154 

.0565(i  .30819 

.95133 

.32474 

.o:$5so  .:!(!  20 

.93999 

3  i 

58 

.27508  .96142 

.20182 

.95647  .30846 

.95124 

.32502 

.91571  .34147 

.93089 

2 

59 

.27536:.  961  34 

.29909 

.9563!)  .30874 

.95115 

.:!2529  .94561  .34175! 

:  9:3970 

1 

60 

.  27564  i.  96126 

.29237 

.95630 

.::0902 

.95106 

.94552 

.84302 

.93969 

0 

- 

Cosin  Sine 

Cosin 

8tne 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine" 

/ 

1 

/ 

74° 

73°       72° 

71° 

70° 

388 


TABLE  X.-SINES  AND  COSINES.  - 


20°   11    21° 

22° 

23° 

24° 

/ 

Sine  Cosin 

Sine  Cosin 

Sine 

Cosin 

Sine  i  Cosin 

Sine  Cosin 

~o 

.34202 

.939(55) 

.35837  .9-3358 

.37461 

.92718 

.39073  .92050 

.40674  .91355  60 

1 

.34229 

.93959 

.35864  .93348 

.37488 

.98707 

.39100  .92039 

.40700  .91343  59 

2 

.34257 

.93949 

.35891  .93337 

.37515 

.92697 

.39127 

.92028 

.40727  .91331  58 

3 

.34284 

.93939 

.35918  .93327 

.37542 

.92686; 

.39153 

.92016 

.40753  .91319  5? 

4 

.34311 

.93929 

.35945  .93316 

.37569 

.92675 

.39180 

.92005 

.40780  .91307  56 

5 

.34339 

.93919 

.35973  .93306 

.37595 

.92664 

.39207 

.91994 

.40806  .91295  55 

6 

.34366 

.93909 

.36000  .93295 

.37622 

.92653 

.39234 

.91982 

.40833  .91283  54 

7 

.34393 

.93899 

.36027  .93285 

.37649 

.92642  .39260 

.91971 

.40860  .91272  53 

8 

.34421 

.93889 

.36054  .93274 

.37676 

.92631 

.39287 

.91959 

.40886  .91260  52 

9 

.34448 

.93879 

.36081  .93264 

.37703 

.92620 

.39314 

.91948 

.40913;.  91248  51 

10 

.34475  .93869 

.36108  .93253 

.37730 

.92609 

.39341 

.91936 

.40939;.  91236  50 

11 

.34503  .93859 

.36135  .93243 

.37757 

.92598 

.39367 

.91925 

.40966 

.91224  49 

12 

.34530  .93849 

.36162  .93232 

.377841.92587 

.39394 

.91914 

.40992 

.91212  48 

13 

.34557 

.9:3839 

.36190  .93222 

.37811 

.92576 

.39421 

.91902 

.41019 

.91200  47 

14 

.34584 

.93829 

.36217 

.93211 

.37838 

.92565 

.39448 

.91891 

.41045 

.91188!  46 

15 

.34612 

.93819 

.36244 

.93201 

.37865 

.92554 

.39474 

.91879 

.41072 

.91176  46 

16 

.34639 

.93809 

.362711.93190 

.37892 

.92543 

.39501 

.91868 

.41098 

.91164  44 

17 

.34666 

.93799 

.36298 

.93180 

.37919 

.92532 

.39528 

.91856 

.41125 

.9115-2  43 

18 

.34694 

.93789 

.36325 

.93169 

.37946 

.92521 

.39555 

.91845 

.41151 

.  91140  j  42 

19 

.34721 

.93779 

.36352  .93159 

.37973  j.92510 

.39581 

.91833 

.41178 

.91128  41 

20 

.34748 

.93769 

.36379 

.93148 

.37999 

.92499 

.39608 

.91822 

.41204 

.91116 

40 

21 

.34775 

.93750 

.36406 

.93137 

.38026 

.92488 

.39635 

.91810 

.41231 

.91104  39 

22 

.34803 

.93748 

.36434 

.93127 

.38053 

.92477 

.39661 

.91799 

.41257 

.91092:  38 

23 

.34830 

.93738 

.36461 

.93116 

.38080 

.92466 

.39688 

.91787 

.41284 

.91080;  37 

24 

.34857 

.93728 

.36488 

.93106 

.38107 

.92455 

.39715 

.91775 

.41310 

.91068  36 

25 

.34884 

.93718 

.  36515  !.  93095 

.38134 

.92444 

.39741 

.91764 

.41337 

.91056  35 

26 

.34912 

.  93708  .  36542  .  93084  .  38161 

.92432 

.39768 

.91752 

.41363 

.91044:  34 

27 

.34939 

.93698 

.36569  .93074 

.38188 

.92421 

.39795 

.91741 

.41390 

.91032  33 

28 

.34966 

.93688 

.36596  .93063 

i  .38215 

.92410 

.39822 

.91729 

.41416 

.91020  32 

29 

.34993 

.93677 

.36623  .93052 

1  .£8241 

.92399 

.39846 

.91718 

.41443 

.91008  31 

30 

.35021 

.93667 

.36650  .93042 

;  .38268 

.92388 

.39875  .01706 

.41469 

.90996 

30 

31 

.35048 

.9365? 

.36677 

.93031 

i  .38295 

.92377 

.39902  .91694 

.41496 

.90984 

29 

32 

.35075 

.93647 

l  .36704 

.93020 

.38322 

.92366  .39928  .91668 

.41522 

.90972 

28 

33 

.35102 

.93637 

.36731  .93010 

.38349 

.92355  .399551.91671 

.41549 

.90960  27 

34 

.35130 

.93626 

.36758  .92999 

.38376 

.92343 

.39982 

.91660 

.41575 

.90948 

26 

35 

.35157 

.93616 

.36785  .92988  !  .38403 

.92332 

!  .40008 

.91648 

.41602 

.90936  25 

36 

.35184 

.93606 

.36812  .92978  .384:30 

.92321 

.40035 

.91636 

.41628 

.909241  24 

37 

.35211 

93596 

.368391.92967 

.38456 

.92310 

.40062 

.91625 

.41655 

.90911  23 

38 
39 

.35239 
.35266 

.93585 
.93575 

.36867 
.36894 

.92956 
.92945 

.38483 
.38510 

.92299  i  .40088 
.92287  1  .40115 

.91613 
.91601 

.41681 
.41707 

.908991  22 
.908871  21 

40 

.35293  .93565 

.36921  .92935 

.38537 

.92276  !  .40141 

.91590 

.41734 

.90875:  20 

41 

.35320  .93555 

.  36948  !.  92924 

.38564 

.92265  .40168 

.91578 

.41760 

.90863  19 

42 

.35347  .93544 

.  36975  !.  92913 

.38591  .92254 

.40195 

.91566 

.417871.90851  18 

43 

.35375  [.93534 

.37002  .92902  .38617 

.92243 

.40221 

.91555 

.41813  .90839  17 

44 

.&5402  .9a524 

.37029  .92892,  .38644 

.92231 

.40248  .91543 

.41840  .90826  16 

45 

.35429 

.93514 

.37'056'.  92881  .38671 

.92220 

.40275 

.91531 

.41866  i.  90814'  15 

46 

.35456 

.93503 

.  37083  !.  92870 

.38698 

.92209 

.40301 

.91519 

.41892  .90802  14 

47 

.35484  .93493 

.37110  .92859 

.38725 

.92198 

.40328  .91508 

.41919  .90790!  13 

48 

.35511  .93483 

.37137).  92849 

.38752 

.92186 

.40355  .91496 

.41945 

.90778  12 

49 

i  35538  !  93472 

.37164  .92838 

.38778 

.92175 

.40381 

.91484 

.41972 

.90766 

11 

50 

.35565 

.93462 

.37191 

.92827 

.38805 

.92164 

.40408  .91472 

.41998 

.90753 

10 

51 

.35592  .93452 

.37218 

.92816 

.38832 

.92152 

.40434  .91461 

.42024 

.90741 

9 

52 

.35619  .93441 

.37245 

.92805 

.38859 

.92141 

.40461  .91449 

.42051 

.90729 

8 

53 

.356471.93431 

.37272 

.92794 

.38886 

.92130 

.40488  .91437 

.42077 

.90717 

7 

54 

.35674  .93420 

.37299 

.92784 

!  .38912 

.92119 

.40514  .91425 

.42104 

.90704 

6 

55 

.  35701  ;.  93410 

.37326 

.92773 

i.  38939  1.92107 

.40541  .91414 

.42130 

.90692 

5 

56 

.35728  .93400 

.37353 

.92762 

;  .38966 

.92096 

.40567  .91402 

.42156  .90680 

4 

57 

.35755  .93389 

.37:380 

.92751 

.38993 

.92085  .40594  .91390 

.42183  .90668 

3 

58 

.35782^.93379 

.37407 

.92740 

.39020 

.92073  .40621  .91378 

.42209L90655 

2 

59 

.35810  .93368 

.37434 

.92729 

.39046 

.92062  .40647  .91366 

.422351.90643 

1 

60 

.35837  .93358; 

.37461 

.92718 

.89078 

.92050  .40674  .91355 

.42262  .90631 

0 

/ 

Cosiu  'Sine  Cosin  j  Sine 

Cosin 

Sine  Cosin  Sine 

Cosin 

Sine 

/ 

69°    ||   68° 

67°   II   66° 

65° 

TABLE  X.— SINES  AND  COSINES. 


25° 

I    26° 

27° 

OOo 

29° 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine  |  Cosin 

Sine  Cosin 

/ 

0 

.422621.90631 

.43837 

.89879 

.45399 

.89101 

.46947  .88295 

.48481  .87462 

60 

1 

,42288|.90618 

.43863 

.89867 

.45425 

.89087  .46973  .88281 

.48506 

.87448 

59 

2 

.423151.90606 

.43889 

.89854 

.45451 

.89074!  .469991.88267 

.48532 

.87434 

58 

3 

.42341 

.90594 

.43916 

.89841 

.45477  .89061  1.470241.88254 

.48557  .87420 

57 

4 

.42367 

.90582 

.43942 

.89828 

.45503 

.89048 

.47050 

.88240 

.48583  .87406 

56 

5 

.42394 

.90569 

.43968 

.89816 

.45529 

.89035 

.47076 

.88226 

.48608 

.87391 

55 

6 

.42420 

.90557 

.43994 

.89803 

.45554 

.89021 

.47101 

.88213 

.48634 

.87377 

54 

7 

.42446 

.90545 

.44020 

.89790 

.45580 

.89008 

.47127 

.88199 

.48659  .87363 

53 

8 

.42473 

.90532 

.44046 

.89777 

.45606 

.88995 

.47153 

.88185 

.48684!.  87349 

52 

9 

.42499 

.90520 

.44072 

.89764 

.45632 

.88981 

.47178 

.88172 

.48710  .87335 

51 

10 

.42525 

.90507 

.44098 

.89752 

.45658 

.88968 

.47204 

.88158 

.48735 

.87321 

50 

11 

.42552 

.90495 

.44124 

.89739 

.45684 

.88955 

.47229 

.88144 

.48761 

.87306 

49 

12 

.42578 

.90483 

.44151 

.89726 

.45710 

.88942 

.47255 

.88130 

.48786 

.87292 

48 

13 

.42604 

.90470 

.44177 

.89713 

.45736 

.88928 

.47281!.  88117 

.48811 

.8727-8 

47 

14 

.42631 

.90458 

.44203 

.89700 

.45762 

.88915 

.47306 

.88103 

.48837 

.87264 

46 

15 

.42657 

.90446 

.44229 

.89687 

.45787 

.88902 

.47-332 

.88089 

.48862 

.87250 

45 

16 

.42683 

.90433 

.44255 

.89674 

.45813 

.88888 

.47358 

.88075 

.48888 

.87235 

44 

17 

.42709 

.90421 

.44281 

.89662 

.45839 

.88875 

.47383 

.88062 

.48913 

.87221 

43 

18 

.42736 

.90408 

.44307 

.89649 

.45865 

.88862 

.47409 

.88048 

.48938 

.87207 

42 

19 

.42762 

.90396 

.44333 

.89636 

.45891 

.88848 

.474341.88034 

.48964 

.87193 

41 

20 

.42788 

.90383 

.44359 

.89623 

.45917 

.88835 

i  .47460 

.88020 

.48989 

.87178 

40 

21 

.42815 

.90371 

.44385 

.89610 

.45942 

.88822 

.47486 

.88006 

.49014 

.87164 

39 

22 

.42841 

.90358 

.44411 

.89597 

.45968 

.88808 

.475111.87993 

.49040 

.87150 

38 

23 

24 

.42867 
.42894 

.90346 
.90334 

.44437 
.44464 

.89584 
.89571 

.45994 
.46020 

.88795 
.88782 

.47537  .87979 
.47562  .87965 

.49065 
.49090 

.87136 
.87121 

37 
36 

25 

.42920 

.90321 

.44490 

.89558 

.46046 

.88768 

.47588 

.87951 

.49116 

.87107 

35 

26 

.42946 

.90309 

.44516 

.89545 

.46072 

.88755 

.47614 

.87937 

.49141 

.87093 

34 

27 

.42972 

.90296 

.  44542  l.  89532 

.46097 

.88741 

.47639 

.87923 

.49166 

.87079 

33 

28 

.42999 

.90284 

.44568  .89519 

.46123 

.88728 

.47665  .87909 

.49192 

.87064 

32 

29 

.43025 

.9)271 

.44594 

.89506 

.46149 

.88715 

.47690 

.87896 

.49217 

.87050 

31 

30 

.43051 

.90259 

.44620 

.89493 

.46175 

.88701 

.47716 

.87882 

.49242 

.87036 

30 

31 

.43077 

.90246 

.44646 

.89480 

.46201 

.88688 

.47741 

.87868 

.49268 

.87081 

29 

32 

.43104 

.90233 

.44672 

.89467 

.46226 

.88674 

.47767 

.87854 

.49293 

.87007 

28 

33 

.43130 

.90221 

.44698  .89454 

.46252 

.88661 

.47793 

.87840 

.49318 

.86993 

27 

34 

.43156 

.90208 

.44724 

.89441 

.46278 

.88647 

.47818 

.87826 

.49344 

.86978 

26 

35 

.43182 

.90196 

.44750 

.89428 

.4630-1 

.88634 

.47844 

.87812 

.49369 

.86964 

25 

36 

.43209 

.90183 

.44776 

.89415 

.46330 

.88620 

.47869 

.87798 

.49394 

.86949 

24 

37 

.43235 

.90171 

.44802 

.89402 

.46355 

.88607 

.47895 

.87784 

.49419 

.86935 

23 

38 

.43261 

.90158 

.44828 

.89389 

.46381  .88593 

.  47920  !.  87770 

.49445 

.86921 

22 

39 

.43287 

.90146 

.44854 

.89376 

.46407  1.88580 

.47946  .87756 

.49470 

.86906 

21 

40 

.43313 

.90133 

.44880 

.89363 

.46433 

.88566 

.47971;.  87743 

.49495 

.86892 

20 

41 

.43340 

.90120 

.44906 

.89350 

.46458 

.88553 

.  47997  !.  87729 

.49521 

.86878 

19 

42 

.43366 

.90108 

.44932 

.89337 

.  46484  !.  88539 

.48022  .87715 

.49546 

.86863 

18 

43 

.43392 

.90095 

.449581.89324 

.46510 

.88526 

.48048  .87701 

.49571 

.86849 

17 

44 

.43418 

.90082 

.  44984  !.  -89311 

.46536 

.88512 

.48073  .87687 

.49596 

.86834 

16 

45 

.43445 

.90070 

.45010  .89298 

.46561 

.88499 

.  48099  ;.  87673 

.49622 

.86820 

15 

46 

.43471 

.90057 

.450361.89285 

.46587 

.88485 

.48124  .87659 

.49647 

.86805 

14 

47 

.43497 

.90045 

.45062 

.89272 

.46613 

.88472 

.48160  .87645 

.4967'2 

.86791 

13 

48 

.43523 

.90032 

.45088 

.89259 

.46639 

.88458 

.481751.87631 

.49697 

.86777 

12 

49 

.43549 

.90019 

.45114 

.89245 

.46664 

.88445 

.48201 

.87817 

.49723 

.86762 

11 

50 

.43575 

.90007 

.45140 

.89232 

.46690 

.88431 

.48226 

.87603 

.49748 

.86748 

10 

51 

.43602 

.89994 

.45166 

.89219 

.46716 

.88417 

.48252 

.87589 

.49773 

.86733 

9 

52 

.43628 

.89981 

.45192 

.89206 

.46742 

.88404 

.48277 

.87575 

.49798 

.86719 

8 

53 

.43654 

.89968 

.45218 

.89193 

.46767 

.88390! 

.48303 

.87561 

.49824 

.86704 

7 

54 

.43680 

.89956 

.45243 

.89180 

.46793 

.88377 

.48328 

.87546 

.49849 

.86690 

6 

55 

.43706 

.89943 

.45269 

.89167 

.46819 

.88363 

.48354 

.87532 

.49874 

.86675 

5 

56 

.43733 

.89930 

.45295 

.89153 

.46844 

.88349 

.48379 

.87518 

.49899 

.86661 

4 

57 

.43759 

.89918 

.45321 

.89140 

.46870 

.88336 

.48405 

.87504 

.49924 

.86646 

3 

58 

.43785 

.89905 

.45347 

.89127 

.46896 

.88322 

.48430 

.87490  .49950 

.86632 

2 

59 

.43811 

.89892 

.45373 

.89114 

.46921 

.88308 

.48456 

.87476 

.49975 

.86617 

1 

60 

.43837 

.89879 

.45399 

.89101 

.46947 

.88295 

.48481  .87462 

.50000 

.86603 

_0 

/ 

Cosin  |  Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin  j  Sine 

Cosin 

Sine 

/ 

64° 

63° 

62° 

61° 

60° 

TABLE  X. -SINES  AND  COSINES. 


30° 

31° 

32°       33° 

34° 

/ 

Sine 

Cosin 

Sine 

Cosin 

Sine  1  Cosin  Sine  !  Cosin 

Sine 

Cosin 

~0 

750000 

.86603 

".51504 

.85717 

752992  :.  84805 

.54464!.  83867 

755919 

."82904 

60 

1 

.50025 

.86588 

.51529 

.85702 

.  53017  !.  84789 

.54488  .83851 

.55943 

.82887 

59 

2 

.50050 

.86573 

.51554 

.85687 

.53041  .84774  .54513  .83835 

.55968 

.82871 

58 

3 

.500761.80559 

.51579 

.85672 

.53066  .84759  .54537  .83819 

.55992 

.82855  57 

4 

.50101 

.86544 

.51604 

.85657 

.53091  .847'43 

.54561 

.83804 

.56016 

.82839;  56 

5 

.50126 

.86530 

.51628 

.85642 

.53115  .84728 

.54586  '.83788 

.56040 

.82822!  55 

6 

.50151 

.86515 

.51653 

.85627 

.53140  .84712 

.54610  .83772 

.56064 

.82806  54 

7 

.50176 

.86501 

.51678 

.85612 

.53164  .84697 

.54635 

.83756 

.56088 

.82790  53 

8 

.50201 

.86486 

.51703 

.85597 

.53189!.  84681^ 

.54659 

.83740 

.56112 

.82773!  52 

9 

.50227 

.86471 

.51728 

.85582 

.532141.84666 

.54683  .83724 

.56136 

.  82757  i  51 

10 

.50252 

.86457 

.51753 

.85567 

.53238  .84650 

.54708 

.83708 

.56160 

.82741 

50 

11 

.50277 

.86442 

.51778 

.85551 

.53263 

.84635 

.54732 

.83692 

.56184 

.82724 

49 

12 

.50302 

.86427 

.51803 

.85536 

.53288 

.84619 

.54756 

.83676 

.56208 

.  827'08'  48 

13 

.50327 

.8(5413 

.51828 

.85521 

.53312  .  84604  : 

.54781 

.83660 

.56232 

.82692 

47 

14 

.50352 

.86398 

.51852 

.85506 

.53337 

.84588. 

.54805 

.83645 

.56256 

.82675 

46 

15 

.50377 

.86384 

.51877 

.85491 

.53361 

.84573] 

.54829 

.83629 

.56280 

.82659  45 

16 

.50403 

.86369 

.51902 

.85476 

.53386 

.84557 

.54854 

.83613 

.56305 

.82643!  44 

17 

.50428 

.86354 

.51927 

.85461 

.53411 

.84542 

.54878 

.83597 

.56329 

.82626  43 

18 

.50453 

.86340 

.51952 

.85446 

.53435 

.84526 

.54902 

.83581 

.56353 

.82610!  42 

19  .50478 

.86325 

.51977 

.85431 

.53460 

.84511 

f>4927 

.83565 

.56377 

.82593  41 

20  .50503 

.86310 

.52002 

.85416 

.53484 

.84495  .54951 

.83549 

.56401 

.82577  40 

21 

.50528 

.86295 

.52026 

.85401 

.53509 

.  84480  •• 

.54975 

.  83533  ! 

.56425 

.82561 

39 

22 

.50553 

.86281 

.52051 

.85385 

.53534 

.84464 

.54999 

.83517 

.56449 

.  82544  !  38 

23 

.50578 

.86266 

.52076 

.85370 

.53558 

.84448 

.55024 

.83501 

.56473 

.82528;  37 

24 

.50603 

.86251 

.52101 

.85355 

.535a3 

.84433 

.55048 

.83485 

.56497 

.82511136 

25 

.50628 

.86237 

.52126 

.85340 

.53607 

.84417 

.55072 

.83469 

.56521 

.82495!  35 

26 

.50654 

.86222 

.52151 

.85325 

.53632 

.84402 

.55097 

.83453 

.56545 

.  82478  34 

27 

.50679 

.86207 

.52175 

.85310 

.53656 

.84386 

.55121 

.83437 

.56569 

.82462!  33 

28 

.50704 

.86192 

.52200 

.85294 

.53681 

.84370 

.55145 

.83421 

.56593 

.82446;  32 

29 

.50729 

.86178 

.52225 

.85279 

.53705 

.84355 

.55169  .83405 

.56617 

.82429  31 

30 

.50754 

.86163 

.52250 

.85264 

.53730 

.84339;  .55194  .83389 

.56641 

.82413  30 

31 

.50779 

.86148 

.52275 

.85249 

.53754 

.84324' 

.55218 

.83373 

.56665 

.82396 

29 

32 

.50804 

.86133 

.52299 

.85234 

.53779 

.84308 

.55242 

.83356 

.56689 

.82380!  28 

33 

.50829 

.86119 

.52324 

.85218 

.53804 

.84292  .55266 

.83340 

.56713 

.82363  27 

34 

.50854 

.86104 

.52349 

.85203 

.53828 

.84277 

.55291 

.83324 

.56736 

.82347  26 

35 
36 

.50879 
.50904 

.86089 
.86074 

.52374 
.52399 

.85188 
.851731 

.53853 

.53877 

.84261 
.84245 

.55315  1.83308 
.55339  .83292 

.56760 
.56784 

.82330 
.82314 

25 
24 

37 

.50929 

.86059 

.52423 

.85157! 

.53902 

.84230 

.55363 

.83276 

.56808 

.82297123 

38 

.50954 

.86045 

.52448 

.85142 

.53926 

.84214 

.55388 

.83260 

.56832 

.82281  22 

39 

.50979 

.86030 

.52473 

.85127: 

.53951 

.84198 

.55412. 

.83244 

.56856 

.82264  21 

40 

.51004 

.86015 

.52498 

.85112 

.53975 

.84182 

.55436 

.83228 

.56880 

.82248 

20 

41 

.51029 

.86000 

.52522 

.85096 

.54000 

.84167 

.55460 

.83212 

.56904 

.82231 

19 

42 

.510541.85985 

.52547 

.85081 

.54024 

.81151 

.55484  .83195 

.56928 

.82214 

18 

43 

.51079  .85970 

.52572 

.85066 

.54049 

.84135 

.  55509  !.  831  79  'I  .56952 

.82198 

17 

44 

.51104 

.85956 

.52597 

.85051 

.54073 

.84120 

.55533  .83163!  .56976 

.82181 

16 

45 

.51129 

.85941 

.52621 

.85035 

.54097 

.84104 

.55557  .83147:  .57000 

.821651  15 

46 

.51154 

.85926 

.52646 

.85020 

.54122 

.84088 

.55581 

.83131  .57024 

.82148  14 

47 

.51179 

.85911 

.53671 

.85005 

.54146 

.84072 

.  55605  .  831  15  i  .  57047  i  .  82132  !  13 

48 

.51204 

.85896 

.52696 

.84989 

.54171 

.84057 

.55630 

.83098  .57071 

.82115  12 

49 

.51229 

.85881 

.52720 

.84974 

.54195  .84041 

.59654 

.83082  .57095 

.82098  11 

50 

.51254 

.85866 

.52745 

.84959 

.542201.84025 

.55678 

.83066  .57119 

.82082  10 

51 

.51279 

.85851 

.52770 

.84943 

.54244  .84009 

.55702 

.83050  .57143 

.82065 

9 

52 

.51304 

.85836 

.52794 

.84928 

.54269  .83994 

.55726 

.83034  .57167 

.82048  8 

53 

.51329 

.85821 

.52819 

.84913 

.542931.83978 

.55750 

.83017!  .57191 

.82032 

7 

54 

.51354 

.85806 

.52844 

.84897 

.  54317  j.  83962 

.55775 

.83001!  .57215 

.82015 

6 

68 

.51379 

.85792 

.52869 

.84882 

.543421.83946 

.55799 

.82985;  .57238 

.81999  5 

56 

.51404 

.85777 

.52893 

.84866 

.54366  .83930 

.55823 

.82969'  .57262 

.81982!  4 

57 

.514291.85762 

.52918 

.84851 

.54391  .83915 

.55847 

.82953  .57286 

.81965!  3 

58 

.51454 

.86747 

.52943 

.84836 

.544151.83899  .55871 

.82936  .57310 

.81949!  2 

59 

.51479 

.85732 

.52967 

.84820 

.54440  .83883  .55895 

.82920  .57&S4 

.  81932  !  1 

60 

.51504 

.85717 

.52992 

.84805 

.54464  .83867 

.55919 

.82904!  .57358 

.81915|  0 

/ 

Cosin 

Sine 

Cosin 

Sine 

Cosin  Sine  Cosin 

"Sine" 

Cosin 

Sine  i 

59° 

58° 

57°   11    56° 

55° 

TABLE  X.-SINES  AND  COSINES. 


/ 

~o 

35°       36° 

37° 

38° 

39° 

i\n 

Sine  !  Cosin 
.57358-.  81915  : 

Sine  Cosin 

758779  .80902 

Sine 

.60182 

Cosin 

79864  : 

Sine  |  Cosin  Sine 
01500  78801  :  ~(i-><i:V> 

Cosin 

~777'1  5 

1  .57381:.  81899 

.58802 

.80885 

.60205  .79846  .61589:  .78783! 

62955 

.77090  59 

2  .57405  .81882 

.58826 

.80867; 

.  60228  .  79829  .6161*  .  78705  !  |  .  62977  1  .  77678  58 

3  .57429  .81865; 

.58849 

.80850: 

.  00251  .  7981  1  ;  .  01  035  .  78747  i  .  63000 

.77660!  57 

4 

.57453  .81848, 

.58873 

.80833 

.60274!.  79793 

.61058 

.78729  .63022 

.77641  50 

5 

.57477  .81832! 

.58896  .80816 

.60298  .79776 

.61681 

.787'lli  .030451.77623!  55 

6 

.57501 

.81815 

.58920  .80799 

.60321 

.79758 

.61704 

.78694 

.63068 

.77005 

54 

7 

.57524 

.81798 

.58943 

.80782 

.60344  .79741 

.61726 

.78676 

.63090 

.77586 

53 

8 

.57548 

.81782 

.58967 

.80765 

.60307 

.79723 

.61749 

.78658 

.63113 

.77568 

52 

9 

.57572 

.81765 

.58990 

.80748 

.60390 

.79706 

.61772 

.78640; 

.63135  .77550 

51 

10 

.57'596 

.81748 

.59014 

.80730 

.60414 

.79688 

.61795 

.78022; 

.63158 

.77531 

50 

11  .57619 

.81731 

.59037 

.80713 

.60437 

.79071 

.61818 

.78004' 

.63180 

.77513 

40 

12 

.57643 

.81714 

.59001 

.80096 

.60460 

.79053 

.618411.78586; 

.03203 

.77494 

48 

13 

.57667 

.81698 

.59084 

.80079 

.60483 

.79635 

.C1864  .78568 

.63225 

.77476 

47 

14 

.57691 

.81681 

.59108 

.80002 

.60506  .79618 

.61887  .78550 

.63248 

.77458 

46 

15  .57715 

.81664 

.59131 

.800441 

.60529 

.79000 

.61909  .78532 

.03271 

.77439 

45 

16  .57738  .81647 

.59154 

.80027 

.60553 

.79583 

.61932  .78514 

.63293 

.77421 

44 

.  17  j  .57762 

.81631 

.59178 

.80610 

.60576 

.79505 

.61955  .78496 

.63316 

.77402  43 

18  .57786 

.81614 

.59201 

.80593 

.60599 

.79547 

.61978  .78478 

.63338 

.77384  42 

19  l  .57810 

.81597 

.59225 

.80576 

.60622 

.79530 

.62001  .78460 

.63361 

.77366  41 

20  .57833 

.81580 

.59248 

.80558 

.60645 

.79512 

.62024  .78442 

.63383 

.77347  40 

21  .57857 

.81563 

.59272 

.80541 

.60668 

.79494 

.62046  .78424 

.63406 

.77329  39 

22  .57881 

.81546 

.59295 

.80524 

.00091 

.79477 

.62069  .78405  .63428 

.77310!  38 

23  .57904 

.81530 

.59318 

.80507, 

.60714 

.79459 

.62092  .78387 

.63451 

.77292  37 

24  .57928 

.81513 

.59342 

.80489 

.60738 

.  79441  1  .62115  .78369 

.63473 

.77273  36 

25  .57952 

.81496 

.59365 

.80472 

.60761 

.79424!  .62138  .78351! 

.63496 

.77255  35 

26 

.57976 

.81479 

.59389 

.80455 

.60784 

.79406i  .62160 

.78333 

.63518 

.77236  34 

27 

.57999 

.81462 

.59412 

.80438 

.60807 

.79388;  .62183  .78315! 

.03540 

.77218  33 

28 

.58023 

.81445 

.59436  .80420 

.60830 

.79371  i  .62206!.  78297  i 

.63563 

.77199  32 

29 

.58047 

.81428 

.594591.80403 

60853 

.79353  .62229  .78279 

.63585 

.77181  31 

30 

.58070 

.81412 

.59482  .80386; 

.60876 

.79335 

.62251 

.78261 

.63608 

.77162 

30 

31 

.58094 

.81395 

.59506  .80368 

.60899 

.79318 

.62274 

.78243 

.63630 

.77144 

29 

32 

.58118  .81378 

.59529  .80351 

.60922 

.79300  .62297 

.  78225  ; 

.63653 

.77125 

28 

33 

.581411.81361 

.59552 

.80334 

.60945 

.79282 

.62320 

.78206 

.63675 

.77107 

27 

34 

.  58165  j.  81344, 

.595761.80316 

.60968 

.79264 

.62342 

.78188 

.63698 

.77088 

26 

35 

.581891.81327 

.59599J.80299 

.60991 

.79247  .62365  .78170  .63720 

.77070 

25 

36 

.58212  .81310 

.59622 

.80282 

.61015 

.79229i!  .62388  .78152  i  .63742 

.77051 

24 

37 

.58236f.81293 

.59646 

.80264! 

.61038 

.  79211  ji.  62411 

.78134  ;  .03705 

.77033 

23 

38 

.  58260  |.  81276 

.59669 

.80247; 

.61061 

.79193  .62433 

.781  16  i  .03787 

.77014 

22 

39 

.58283L  81259 

.59693 

.80230 

.61084 

.79176  .62456 

.780981 

.03810 

.76996  21 

40 

.58307!.  81242 

.59716 

.80212 

.61107 

.79158;  .62479 

.78079; 

.63832 

.76977 

20 

41 

.58330  .81225 

.59739 

.80195 

.61130 

.79140 

.62502 

.  78061  ' 

.63854 

.76959 

19 

42  !.  58354  i.  81208 

.59763 

.  80178  ; 

.61153 

.79122  .625241.78043 

.63877 

.70940!  18 

43 

.58378|.81191 

.59786 

.80160 

.61176  .79105 

.  62547  i.  78025 

.63899 

.70921  17 

44 

.584011.81174 

.59809 

.80143 

.61199!.  7  9C87 

.62570 

.78007 

.63922 

.76903!  16 

45 

.584251.81157 

.59832 

.80125 

.61222;.  79069 

.62592 

.77988 

.63944 

.76884!  15 

46 

.58449;.  81  140 

.59856 

.80108 

.61245).  79051 

.62015 

.77970 

.03966 

.76866  14 

47 

.584721.81123 

.59879 

.80091 

.61208  .79033 

.62038  .77952 

.63989 

.76847  13 

48 

.58496 

.81106 

.59902 

.80073 

.61291  .79016 

.62000!.  77934 

.640111.76828  12 

49 

.58519 

.81089 

.59926 

.  80056  i 

.61314!.  78998 

.62683!.  77916 

.64033 

.76810  11 

50 

.58543 

.81072 

.59949 

.80038 

.61337 

.78980 

.62706 

.77897 

.64056 

.76791  j  10 

51 

.58567 

.81055 

.59972 

.80021 

.61360 

.78962  !  .62728 

.77879 

.64078 

.76772  9 

52 

.68590 

.81038 

.59995 

80003 

.61383 

.78944  .62751 

.77861 

.64100 

.70754:  -8 

53 

.58614 

.81021 

.60019 

.79986 

.61406 

.78926 

.62774 

.77843' 

.64123 

.70735!  7 

54 

.58637 

.81004 

.60042 

.79968 

.61429 

.78908 

.62796 

.77824 

.64145 

.79717  6 

55 

.58001 

.80987 

.00005 

.79951 

.61451 

.78891 

.62819 

.77806 

.64167 

.  76698  5 

56 

.58684 

.80970 

.60089 

.79934 

.61474 

.78873 

.62842 

.77788 

.64190 

.76679!  4 

57 

.587'08 

.80953 

.60112 

.79916 

.61497 

.78855  j  .62864 

.77709 

.64212 

.76001   3 

58  !  .58731 

.80936 

.60135 

.79899 

.61520 

.78837  ;  .62887 

.77751 

.64234 

.76642!  2 

59  !  .58755 

.80919 

.60158  .79881  .61543 

.78819 

.02909 

.777-33 

.64256 

.76623  1 

00  .58779 

.80902 

.60182  .79804 

.61566 

.78801 

.02932 

.77715 

.64279 

.70001 

0 

Cosin 

Sine 

Cosin  Sine 

Cosin  Sine 

Cosin 

"Sine 

Cosin 

Sine 

i  1  j  .  . 

54° 

53°       52° 

51° 

50° 

TABLE   X.— SINES    A.ND 


40°    i!    41°    ||    42° 

43°    i 

44° 

i  \  
Sine 

Cosin  '  Sine 

Cosin  '  i  Sine  Cosin 

Sine  ;  Cosin 

Sine 

Cosin 

0 

.64279 

.76004  .65600 

.75471 

.00913 

.74314 

.68200 

.73135 

.69466 

.71934 

60 

1 

64301 

.  70586 

.65628 

.75452 

.66935 

.74295 

.68221 

.73116 

.69487 

.71914 

59 

2 

.64323 

.76567 

.65650 

.75433 

.66956 

.74276 

.68242 

.73096 

.69508 

.71894  58 

3 

'  043  16 

.76548 

.65672 

.75414  .66978 

.74256 

.68264 

.73076 

.69529 

.71873 

57 

4 

.64308 

70530 

.65694 

.75395  .66999 

.74237 

.68285 

.73056 

.69549 

.71853 

56 

5 

.  04390 

.76511 

65716 

.75375 

.67021 

.74217 

.08306 

.73036 

.6957'0 

.71833 

55 

6 

'64412 

!  76492 

.65738 

.  75356  i 

.67043 

.74198 

.68327 

.73016 

.695911.71813 

54 

7 

.044351.76473 

.65759 

.753371 

.67064 

.74178 

.68349 

.72996 

.69612 

.71792 

53 

8 

.04457 

76455 

.65781 

.75318 

.67086 

.74159 

.68370 

.72976 

.69633 

.71772 

52 

g 

.04179 

7iU3R 

.65803 

.75299 

.67107 

.74139 

.68391 

.72957 

.69654 

.71752 

51 

10 

.04501;.  76417 

.65825 

.75280! 

.67129 

.74120 

.68412 

.72937 

.69675 

.71732 

50 

11 

.04524 

.76398 

.65847 

.75261 

.67151 

.74100 

.68434 

.72917 

.69696 

.71711 

49 

12 

.04546 

70330 

.65309 

.75241 

.67172 

.74080 

.68455  '.72897 

.69717 

.71691 

48 

13 

.64508 

.70301 

.65891 

.75232 

.67194 

.74061 

.68476  .72877 

.69737 

.71671 

47 

It 

.64590 

70342  .65913 

.75203 

.67215 

.74041 

.68497 

.72857 

.69758 

.71650 

46 

15 

.64612 

.76323 

.65935 

.75184 

.67237 

.74022 

.68518 

.72837 

.69779 

:  71630 

45 

16 

.64635 

.  76304  • 

.65956 

.75165 

.67258 

.74002 

.68539 

.72817 

.69800 

.71610 

44 

17 

.64057  .70236  .65978 

.75146 

.67280 

.73983 

.68561 

.72797 

.69821 

.71590 

43 

18 

.04679 

.76267  .66000 

.75126 

.67301 

.73963 

.68582 

72777 

.69842 

.71569 

42 

19 

.64701 

.76248  .60022 

.75107 

.67323 

.73944 

.68603 

!  72757 

.69862 

.71549 

41 

20 

.64723 

.76229,  .66044 

.75088 

.67344 

.73924 

.68624 

.72737 

.69883 

.71529 

40 

21 

.64746 

.76210  '  L66066 

.75069 

.67366 

.73904 

.68645 

.72717 

.69904 

.71508 

39 

22 

.64768 

.76192  .66033 

.  75050  j 

.67387 

.73885 

.68606 

.72697 

.69925 

.71488 

38 

.64790 

.761731 

.68109 

.75030! 

.67409 

.73865 

.68688 

.72677 

.69940 

.71468!  37 

24 

.64812 

.  76154  ' 

.66131 

.75011! 

.67430 

.73846 

.687-09 

.72657 

.69966 

.71447 

36 

25  .61834 

.76135 

.66155 

.74932 

.67452 

.78826 

.68730 

.72037 

.69987 

.71427 

35 

26  .64856 

.76116 

.66175 

.74973 

.67473 

.73806 

.68751 

.72617 

.70008 

.71407 

34 

27  .04S78 

.76097 

.66197 

.  74953  i 

.67495 

.73787 

.68772 

.72597 

.70029 

.71386 

33 

28  .64901 

.76078 

.6(3218 

,74034 

-.67516 

.73767 

.68793 

.72577| 

.70049 

.71306 

32 

29  .64923 

.76059 

.66240 

.74915 

.67538 

.73747 

.68814 

.72557 

.70070 

.71345 

31 

30  .64945 

.76041 

.66262 

.74896;  .67559 

.73728 

.68835 

.  72537  j 

.70091 

.71325 

30 

31  .64967 

.76022! 

.66284 

74876  .67580 

.73708 

.68857 

.72517; 

.70112 

.71305 

29 

32  .64989 

.76003  i  .66:306 

74857  .67602 

.73688 

.68878 

.72497, 

70132 

.71284 

28 

33  .65011 

.75984  .66327 

74333  .67623 

.73609 

.68899 

.72477, 

.70153 

.71264 

27 

34  .65033 

.75965  .66349 

74818  .07645 

.73649 

.68920 

.72457 

.70174 

.71243 

26 

35  .65055 

.75946 

.66371 

74799 

.67666 

.73629 

.68941 

.72437 

.70195 

.71223 

35 

36  .65077 

.75927 

.66393 

74780  .67688 

.73610 

.68962 

.7241? 

.7C215 

.71203 

24 

37  .65100 

.75903  .66414 

.74760  .67709 

.73590 

.68983 

.72397 

.70236 

.71182 

23 

38  .65122 

.75889 

.66436 

74741 

.67730 

73570 

.69004 

.72377 

.7U257 

.71162 

22 

39 

.65144 

.75870 

.68458 

74722  .67752 

.73551 

.69025 

.72357 

.70277 

.71141 

21 

40 

.65166 

.75851 

.66480 

.74703; 

.67773 

.73531 

.69046 

.72337 

.70298 

.71121 

20 

41 

.65188 

.75832 

.66501 

.74683 

.67795 

.73511 

.69067 

.72317 

.70319 

.71100 

19 

42 

.65210 

.75813 

.66523 

.74004  .67816 

.73491 

.69088 

.72297 

.70339 

.71080 

18 

43 

.65232 

.75794 

.68545 

.74644  .67837 

.73472 

.69109 

.72277 

.70360 

.710.M)  17 

44 

.65254 

.75775 

.66566 

.74625  .67859 

.73452 

.69130 

.72257 

.70381 

.71039!  16 

45 

.65276 

.75756 

.665881.74606  .67880 

.73432 

.69151 

.72236 

.70401 

.710191  15 

46 

.65298 

.75738 

.60610  .74586  .r,79itl 

.73413 

.69172 

.72216 

.70422 

.70998!  14 

47 

.65320 

.75719 

.66632  '.74567  !  .67923 

.73393 

.69193 

.72196  i  .70443 

.70978  13 

48 

.65342 

.75700 

.66653  .74548  .67944 

.73373 

.69214 

.72176 

.70465 

.709571  12 

49  .65364 

.75680 

.66675  .74523  .67965 

.73353 

.69235 

.72156 

.70484 

.70937;  11 

50 

.65386 

.75661; 

.666971.74509  .079S7 

.73333  .69256 

.72136 

.70505 

.70916  10 

51 

.65408 

.75642 

.66718  .74489  .68008 

.73314  T69277 

.72116 

.70525 

.70896  9 

52 

.65430 

.75623;  .66740  .74470  .68029 

.73294  !  .69298  .72095 

.70546 

.70875'  8 

53 
54 

.65452  .75604 
.65474!.  75585 

.66762  .74451  ;  .68051 
.66783  .74431  .68072 

.73274 

.73254 

.  69319  j.  72075 
.69340  .72055 

.70567 

.70587 

.70855:  7 
.70834  6 

55 

.65496 

.75566 

.66805  .74412  !  .68093 

.73234 

.69361 

.72035 

.70608 

.  70813  i  5 

56 

.65518 

.75547: 

.66827  .74392  .68115 

.73215 

.69382 

.72015 

.70628 

.70793  4 

57 

.65540 

.75528 

.66848  .74373  .68136 

.73195 

.69403 

.71995 

.70649 

.70772  3 

58 

.65562 

.75509 

.66870  .74353  .68157 

.73175 

j.69424 

.71974 

.70670 

.70752  2 

59 

.65584 

.75490 

.66891  .74834  .68179 

.73155 

i  .69445  .71954 

.70690 

.70731   1 

60 

.65606 

.75471 

.66913  .74314: 

.68200 

.73135 

.69466 

.  71934  .70711 

.70711  0 

/ 

Cosin 

Sine 

Cosin  Sine 

Cosin 

Sine 

Cosin  Sine  :  Cosin 

Sine 

f 

49° 

48°       47°       46°       45° 

293 


TABLE  XL— SECANTS  AND  COSECANTS. 


SECANTS. 

' 

0° 

1° 

2° 

3° 

4° 

5° 

t 

0 

1-0000000 

1-0001523 

1-0006095 

1-0013723 

1-0024419 

1-0038198 

60 

1 

1-0000000 

1-0001574 

1-0006198 

1-0013877 

1-0024623 

1-0038454  1  59 

2 

1-0000002 

1  -0001627 

1-0006300 

1-0014030 

1-0024829 

1-0038711   58 

3 

1-0000004 

1-0001679 

1-0006404 

1-0014185 

1-0025035 

1-0038969 

57 

4 

1-0000007 

1-0001733 

1-0006509 

1-0014341 

1-0025241 

1-0039227 

56 

5 

1-0000011 

1-0001788 

1-0006614 

1-0014497 

1-0025449 

1-0039486 

55 

6 

1-0000015 

1-0001843 

1-0006721 

1-0014655 

1-0025658 

1-0039747 

54 

7 

l-00000i>l 

1-0001900 

1-0006828 

1-0014813 

1-0025867 

1-0040008 

53 

8 

1-0000027 

1-0001957 

1-0006936 

1-0014972 

1-0026078 

1-0040270 

62 

9 

1-0000034 

1-0002015 

1-0007045 

1-0015132 

1-0026289 

1-0040533 

51 

10 

1-0000042 

1-0002073 

1-0007154 

1-0015293 

1-0026501 

1-0040796 

50 

11 

1-0000051 

1-0002133 

1-0007265 

1-0015454 

1-0026714 

1-0041061 

49 

12 

1-0000061 

1-0002194 

1-0007376 

1-0015617 

1-0026928 

1-0041326 

48 

13 

1-0000072 

1-0002255 

1-0007489 

1-0015780 

1-0027142 

1-0041592 

47 

14 

1-0000083 

1-0002317 

1-0007602 

1-0015944 

1-0027358 

1-0041859 

46 

15 

1-0000095 

1-0002380 

1-0007716 

1-0016109 

1-0027574 

1-0042127 

45 

16 

1-0000108 

1-0002444 

I1  0007830 

1-0016275 

1-0027791 

1-0042396 

44 

17 

1-0000122 

1-0002509 

1-0007946 

1-0016442 

1-0028009 

1-0042666 

43 

18 

1-0000137 

1-0002575 

1-0008063 

1-0010609 

1-0028228 

1-0042937 

42 

19 

1-0000153 

1-0002641 

1-0008180 

1-0016778 

1-0028448 

1-0043208 

41 

20 

1-0000169 

1-0002708 

1-0008298 

1-0016947 

1-0028669 

1-0043480 

40 

21 

1-0000187 

1-0002776 

1-0008417 

1-0017117 

1-0028890 

1-0043753 

39 

22 

1-0000205 

1-0002845 

1-0008537 

1-0017288 

1-0029112 

1-0044028 

38 

23 

1-0000224 

1-0002915 

1-0008658 

1-0017460 

1-0029336 

1-0044302 

37 

24 

1-0000244 

1-0002986 

1-0008779 

1-0017633 

1-0029560 

1-0044578 

36 

25 

1-0000264 

1-0003058 

1-0008902 

1-0017806 

1-0029785 

1-0044855 

35 

26 

1-0000286 

1-0003130 

1-0009025 

1-0017981 

1-0030010 

1-0045132 

34 

27 

1-0000308 

1-0003203 

1-0009149 

1-0018156 

1-0030237 

1-0045411 

33 

28 

1-0000332 

1-0003277 

1-0009274 

1-0018332 

1-0030464 

1-0045690 

32 

29 

1-0000356 

1-0003352 

1-0009400 

1-0018509 

1-0030693 

1-0045970 

31 

30 

1-0000381 

1-0003428 

1-0009527 

1-0018687 

1-0030922 

1-0040251 

30 

31 

1-0000407 

1-0003505 

1-0009654 

1-0018866 

1-0031152 

1-0046533 

29 

32 

1-0000433 

1-0003582 

1-0009783 

1-0019045 

1-0031383 

1-0046815 

28 

33 

1-0000461 

1-0003660 

1-0009912 

1-0019225 

1-0031615 

1-0047099 

27 

34 

1-0000489 

1-0003739 

1-0010042 

1-0019407 

1-0031847 

1-0047383 

26 

35 

1-0000518 

1-0003820 

1-0010173 

1-0019589 

1-0032081 

1-0047669 

25 

36 

1-0000548 

1-0003900 

1-0010305 

1-0019772 

1-0032315 

1-0047955 

24 

37 

1-0000579 

1-0003982 

1-0010438 

1-0019956 

1-0032551 

1-0048242 

23 

38 

1-0000611 

1-0004065 

1-0010571 

1-0020140 

1-0032787 

1-0048530 

22 

39 

1-0000644 

1-0004148 

1-0010705 

1-0020326 

1-0033024 

1-0048819 

21 

40 

1-0000677 

1-0004232 

1-0010841 

1-0020512 

1-0033261 

1-0049108 

41 

1-0000711 

1-0004317 

1-0010977 

1-0020699 

1-0033500 

1-0049399 

19 

42 

1-0000746 

1-0004403 

1-0011114 

1-0020887 

1-0033740 

1-0049690 

18 

43 

1-0000782 

1-00044SK) 

1-0011251 

1-0021076 

1-00339SO 

1-0049982 

17 

44 

1-0000819 

1-0004578 

1-0011390 

1-0021266 

1-0034221 

1-0050275 

IS 

45 

1-0000857 

1-0004066 

1-0011529 

1-0021457 

1-0034463 

1-0050569 

15 

46 

1-0000895 

1-0004756 

1-0011670 

1-0021648 

1-0034706 

1-0050864 

14 

47 

1-0000935 

1-0004846 

1-0011811 

1-0021841 

1-0034950 

1-0051160 

13 

48 

1-0000975 

1-0004937 

1-0011953 

1-0022034 

1-0035195 

1-0051456 

12 

49 

1-0001016 

1-0005029 

1-0012096 

1-0022228 

1-0035440 

1-0051754 

11 

50 

1-0001058 

1-0005121 

1-0012239 

1-0022423 

1-0035687 

1-0052052 

10 

51 

1-0001101 

1-0005215 

1-0012384 

1-0022619 

1-0035934 

1-0052351 

9 

52 

1-0001144 

1-0005309 

1-0012529 

1-0022815 

1-0036182 

1-0052651 

8 

53 

1-0001189 

1-0005405 

1-0012670 

1-00-23013 

1-0036431 

1-0052952 

7 

54 

1-0001234 

1-0005501 

1-0012323 

1-00-23211 

1-0036681 

1-0053254 

6 

55 

1-0001280 

1-0005598 

1-0012971 

1-00-23410 

1-0036932 

1-0053557 

5 

56 

1-0001327 

1-0005696 

1-0013120 

1-0023610 

1-0037183 

1-0053860 

4 

57 

1-0001375 

1-0005794 

1-0013269 

1-0023811 

1-0037436 

1-0054164 

3 

58 

1-000142;} 

1-0005894 

1-0013420 

1-0024013 

1-0037689 

1-0054470 

2 

69 

1-0001473 

1-0005994 

1-0013571 

1-0024216 

1-0037943 

1-0054776 

1 

60 

1-0001523 

1-OOOG095 

1-0013723 

1-0024419 

1-0038198 

1-0055083 

0 

i 

89° 

88° 

87° 

86° 

85° 

84° 

' 

COSECANTS. 

TABLE  XI.— SECANTS  AND  COSECANTS. 


SECANTS. 

/ 

6° 

7° 

8° 

9° 

10° 

11° 

t 

o 

1-0075098 

1-6098276 

1-0124651 

1-0154266 

1-0187167 

60 

i 

1-0055083 

1-0075159 

1-0098689 

1-0125113 

1-0154787 

1-0187743 

59 

2 

1-0055391 

1-0075820 

1-0099103 

1-0125586 

1-0155310 

1-0188321 

58 

3 

1-0055099 

1-0070182 

1-0099518 

1-0126055 

1-0155833 

1-0188899 

57 

4 

1-0056009 

1-0076545 

1-0099934 

1-0126524 

1-0156357 

1-0189478 

56 

5 

1-005C319 
1-005G631 

1  -0078908 

1-0100351 

1-0120993 

1-015G882 

1-0190059 

55 

6 

1-0077273 

1-0100769 

1-0127466 

1-0157408 

1-0190640 

54 

7 

1-0056943 

1-007/639 

1-0101187 

1-0127939 

1-0157934 

1-0191222 

53 

8 

1-0057256 

1-0078005 

1-0101607 

1-0128412 

1-0158462 

1-0191805 

52 

9 

1-0057570 

1-0078372 

1-0102027 

1-01*8880 

1-0158991 

1-0192389 

51 

10 

1-0057885 
1-0058200 

1-0078741 

1-0102449 

1-0129361 

1-0159520 

1-0192973 

50 

11 

1  'fin^Qfn? 

1-0079110 

1-0102871 

1-0129837 

1-0160050 

1-0193559 

49 

12 

J.  UUOoOl/ 

1  '0058834 

1-0079480 

1-0103294 

1-0130314 

1-0160582 

1-0194146 

48 

13 

1  '0059153 

1-0079851 

1-0103718 

1-0130791 

1-0161114 

1-0194734 

47 

14 

l'00.r)9472 

1-OOS0222 

1-0104143 

1-0131270 

1-0161647 

1-0195322 

46 

15 

1-0059792 

1-0080595 

1-0104568 

1-0131750 

1-0162181 

1-0195912 

45 

16 

1  '0000113 

1-0080968 

1-0104995 

1-0132230 

1-0162716 

1-0196502 

44 

17 

1  -0000435 

1-0081343 

1-0105422 

1-0132711 

1-0163252 

1-0197093 

43 

18 

1-0060757 

1-OU81718 

1-0105851 

1-0133194 

1-0163789 

1-01976S6 

42 

19 

1-0061081 

1-0082094 

1-0106-280 

1-0133677 

1-0164327 

1-0198279 

41 

20 

1-0061405 

1-0082471 

1-0106710 

1-0134161 

r-0161865 

1-0198873 

40 

21 

1-0061731 

1-0082S49 

1-0107141 

1-0134646 

1-0165405 

1-0199468 

39 

22 

1-0062057 

1-0083228 

1-0107573 

1-0135132 

1-0165946 

1-0200064 

38 

23 

1-006-2384 

1-0083607 

1-0108006 

1-0135618 

1-0166487 

1-0200661 

37 

24 

1-0002712 

1-0083988 

1-0108440 

1-0136106 

1-0167029 

1-0201259 

30 

25 

1-0063040 

1-0084369 

1-0108875 

1-0136595 

1-0167573 

1-0201858 

35 

26 

1-0063370 

1-0084752 

1-0109310 

1-0137084 

1-0168117 

1-0202457 

34 

27 

1-0063701 

1-0085135 

1-0109747 

1-0137574 

1-0168662 

1-0203058 

33 

28 

1-0064032 

1-0085519 

1-0110184 

1-0138006 

1-0169208 

1-020.3660 

32 

29 

1-0064364 

1-0085904 

1-0110622 

1-0138558 

1-0169755 

1-0204262 

31 

30 

1-0064697 

1-0086290 

1-0111061 

1-0139051 

1-0170303 

1-0204866 

30 

31 

1-0005031 

1-0086676 

1-0111501 

1-0139545 

1-0170851 

1-0205470 

29 

32 

1-0065366 

1-0087064 

1-0111942 

1-0140040 

1-0171401 

1-0206075 

28 

33 

1-0065702 

1-0087452 

1-0112384 

1-0140536 

1-0171952 

1  '0200082 

27 

34 

1-0066039 

1-0087842 

1-0112827 

1-0141032 

1-0172503 

1-0207289 

26 

35 

1-0066376 

1-0088232 

1-0113270 

1-0141530 

1-0173056 

1  -0-207897 

25 

36 

1-0066714 

1-0088623 

1-0113755 

1-0142029 

1-0173609 

1-0208506 

24 

37 

1-0067054 

1'00»9015 

1-0114100 

1-0142528 

1-0174163 

1-0209116 

23 

38 

1-0007394 

1-0089408 

1-01H60G 

1-01430-28 

1-0174719 

1-0209727 

23 

39 

1-0067735 

1-0089802 

1-0115054 

1-0143530 

1-0175275 

1-0210339 

21 

40 

1-006S077 

1-0090196 

1-0115502 

1-0144032 

1-0175832 

1-0210952 

20' 

41 

1-0068419 

1-0090592 

1-0115951 

1-0144535 

1-0176390 

1-0211566 

19 

42 

1-0063763 

1-0090088 

1-0116(00 

1-1145039 

1-0176949 

1-0212180 

18 

43 

1  "0069108 

1-0001386 

1-0116851 

1-014554  4 

110177509 

1-0212796 

17 

44 

1  -0069453 
l  .fwo7aa 

1-0091784 

1-0117303 

1-0146050 

1-0178069 

1-0213413 

16 

45 

i  uuowyy 

1-0092183 

1-0117755 

1-0146556 

1-0178081 

1-0214030 

15 

46 

1*0070146 
1  '0070494 

1-0092583 

1-0118209 

1-0147064 

1-0179194 

1-0214649 

14 

47 

1  "0070843 

l-0092l»84 

1-0118663 

1-0147572 

1-0179757 

1-0215208 

13 

43 

1  "0071193 

1-0093386 

1-0119118 

1-0148082 

1-0180321 

1-0215888 

12 

49 

1  "007  15  44 

1-0093788 

1-0119575 

1-0  14859* 

1-0180887 

1-0216510 

11 

50 

1-0094192 

1-0120032 

1-0149103 

1-0181453 

1-0217132 

10 

51 

1-0071895 
1-0072248 

1-0094596 

1-0120489 

1-0149616 

1-0182020 

1-0217755 

9 

02 

1-0072601 

1-00951,01 

1-0120948 

1-0150129 

1-0182588 

1-0218379 

8 

03 

1-0072955 

1-0095408 

1-0121408 

1-01506J3 

1-0183158 

1-0219004 

7 

04 

1-0073310 

1-0095815 

1-0121869 

1-0151158 

1-0183728 

1-0219630 

6 

55 

1-0096223 

1-0122330 

1-0151673 

1-0184298 

l-0220-'57 

5 

1-0073666 

50 

1-0074023 

1-0096631 

1-0122793 

1-0152190 

1-0184870 

1  -0220865 

4 

07 

1-00743SO 

1-0097041 

1-0123206 

1-0152708 

1-0185443 

1-0221514 

3 

08 

1-0074739 

1-0097452 

1-0123720 

1-0153226 

10180017 

1-0222144 

2 

09 

1-0075098 

1-0097863 

1-0124185 

1-0153746 

1-0186591 

1-0222774 

1 

CO 

1-0098276 

1-0124651 

1-0154266 

1-0187167 

1-0223  406 

0 

/ 

83° 

82° 

81° 

80° 

79° 

78° 

t 

COSECANTS. 

TABLE  XI.— SECANTS  AND  COSECANTS. 


SECANTS. 

' 

12° 

13° 

14° 

15° 

16° 

17° 

i  > 

1-0223406 

1-0263041 

1-0306136 

1-0352762 

1-0402994 

1-045R918 

60 

0 

1-0224039 

1-0263731 

1-0306884 

1-0353569 

1-0403863 

1-0457848 

59 

1 

1-0224672 

1-0264421 

1-0307633 

1-0354373 

1-0404732 

1-0458780 

58 

2 

1-0225307 

1-0265113 

1-0308383 

1-0305187 

1-0405602 

1-0459712 

57 

3 

1-0225942 

1-0265806 

1-0309134 

1-0355998 

1-0406473 

1-0460646 

56 

4 
5 

1-0226573 

1-0266499 

1-0309886 

1-0356809 

1-0407346 

1-0461581 

55    1 

g 

1-0227216 

1-0267194 

1-0310639 

1-0357621 

,  1-0408219 

1-0462516 

54 

1-0227854 

1-0267889 

1-0311393 

1-0358435 

1-0409094 

1-0463453 

53 

g 

1-0228493 

1-0268586 

1-0312147 

1-0359249 

1-0409969 

1-0464391 

52 

9 

1-0229133 

1-0269283 

1-0312903 

1-0360065 

1-0410845 

1-0465330 

51 

10 

1-0229774 

1-0269902 

1-0313660 

1-0360881 

1-0411723 

1-0466270 

50 

1_ 

1-0230416 

1-02706S1 

1-0314418 

1-0361699 

1-0412601 

1-0467211 

49 

12 

1-0231059 

1-0271381 

1-0315177 

1-0362517 

1-0413481 

1-046^153 

48 

13 

1-0231703 

1-0272082 

1-0315936 

1-0363337 

1-0414362 

l-046i'096 

47 

14 

1-0232348 

1-0272785 

1-03  16U97 

1-0364157 

1-0415243 

1-0470040 

46 

15 

1-0232994 

1-0273488 

1-0317459 

1-0364979 

1-0416126 

1-0470986 

45 

16 

1-0233641 

1-0274192 

1-0318222 

1-0365SOI 

1-0417009 

1-0471932 

44 

17 

1-0234288 

1-0274897 

1-0318US5 

1-0366625 

1-0417894 

1-0472879 

43 

18 

1-0234937 

1-0275603 

1-0319750 

1-0367449 

1-0418780 

1-0473823 

42 

19 

1-0235587 

1-0276310 

1-0320516 

1-0368275. 

1-0419667 

1-0474777 

41    j 

20 

1-0236237 

1-0277018 

1-0321282 

1-0369101 

1-0420554 

1-0475728 

40    i 

21 

10236889 

1-0277727 

1-0322059 

1-0369929 

1-0421443 

1-0476679 

39 

22 

1-0237541 

1-0278437 

1-03-22818 

1-0370757 

1-0422333 

1-0477632 

38 

23 

1-0238195 

1-0279148 

1-0323588 

1-0371587 

1-0423224 

1-0478586 

37 

24 

1-0238849 

1-0279860 

1-0324359 

1-0372417 

1-0424116 

1-0479540 

36 

25 

1-0239304 

1-0280573 

1-0325130 

1-0373249 

1-0425009 

1-0480496 

35 

26 

1-0240161 

1-0281287 

1-0325903 

1-0374082 

1-0425903 

1-0481453 

34 

27 

1-0240818 

1-02S20U2 

1-0326676 

1-0374915 

1-0426798 

1-0482411 

33 

28 

1-0241476 

1-0282717 

1-0327451 

1-0375750 

1-0427694 

1-0483370 

32 

29 

1-0242135 

1-0283434 

l-032t>227 

1-0376585 

1-0428591 

1-0484330 

31 

30 

1-0242795 

1-0284152 

1-0329003 

1-0377422 

1-0429489 

1-0485291 

30 

31 

1-0243456 

1-0284871 

1-0329781 

1-0378260 

1-0430388 

1-0486253 

29 

32 

1-0244118 

1-0285590 

1-0330559 

1-0379098 

1-0431289 

1-0487217 

28 

33 

1-0244781 

1-0286311 

1-0331339 

1-0379938 

1-0432190 

1-0488181 

27 

34 

1-0245445 

1-0287033 

1-0332119 

1-0380779 

1-0433092 

1-0489146 

26 

35 

1-0246110 

1-0287755 

1-0332901 

1-0381621 

1-0433995 

1-0490113 

25 

36 

1-0246776 

1-0288479 

1-0333683 

1-0382463 

1-0434900 

1-0491080 

24 

37 

1-0247442 

1-0289203 

1-0334467 

1-0383307 

1-0435805 

1-0492049 

23 

38 

1-0248110 

1-0289929 

1-0335251 

1-0384152 

1-0436712 

1-0493019 

22 

39 

1-0248779 

1  -0290655 

1-0336037 

1-0384998 

1-0437619 

1-0493989 

21 

40 

1-0249448 

1-0291383 

1-0336823 

1-0385844 

1-0438528 

1-0494961 

20 

41 

1-0250119 

1-0292111 

1-0337611 

1-0386692 

1-0439437 

1-0495934 

19 

42 

1-0250790 

1-0292840 

1-0338399 

1-0387541 

1-Q440348 

1-0496908 

18 

43 

1-0251463 

1-0293571 

1-03391S8 

1-0388391 

1-0441259 

1-0497883 

17 

44 

1-0252136 

1-0-29430-2 

1-0339979 

1-0389242 

1-0442172 

1-0498859 

16 

45 

1-0252811 

1-0295034 

1-0340770 

1-0390094 

1-0443086 

1-0499836 

15 

48 

1-0253486 

1-02957G3 

1-0341563 

1-0390947 

1-0444001 

1-0500815 

14 

47 

1-0254162 

1-0296502 

1-0342356 

1-0391800 

1-0444917 

1-0501794 

13 

48 
49 

1-0254839 

1-0297237 

1-0343151 

1-0392655 

1-0445833 

1-0302774 

12 

1-0255518 

1-0297973 

1-0343946 

1-0393511 

1-0446751 

1-0503756 

11 

50 

1-0256197 

1-0298711 

1-0344743 

1-0394368 

1-0447670 

1-0504738 

10 

51 

52 

1-0256877 

1-0299449 

1-0345540 

1-0395226 

1-0448590 

1-0505722 

9 

53 

1-0257558 

1-0300188 

1-0346338 

1-0396085 

1-0449511 

1-050«706 

8 

54 

1-0258240 

i-0300928 

1-0347138 

1-0396945 

1-0450433 

1-0507692 

7 

55 

1-025S923 

1-0301669 

1-0347938 

1-0397806 

1-0451357 

1-0508679 

6 

1-0259607 

1-0302411 

1-0348740 

1-0398669 

1-0452281 

1-0509667 

5 

56 
57 

1-0260292 

1-0303154 

1-0349542 

1-039953-2 

1-0453206 

1-0510656 

4 

58 

1-0260978 

1-0303898 

1-0350346 

1-0400396 

1-045413-2 

1-0511646 

3 

59 

1-0261665 

1-0304C43 

1-0351150 

1-0401261 

1-0455060 

1-0512637 

2 

60 

1-026235-2 

1-0305389 

1:0351955 

1-0402127 

1-0455988 

1-0513629 

1 

1-0263011 

1-0306138 

1-0302762 

1-0402994 

1-0456918 

1-0514622 

0 

' 

77° 

76° 

75° 

74° 

73° 

72° 

' 

COSECANTS. 

SECANTS. 

18° 

19° 

20° 

21° 

22° 

23° 

i 

0 

1-0514622 

1-0576207 

1-0641778 

1-0711450 

1-0785317 

1-OB63604 

GO 

1 

1-0515617 

1-0577267 

1-064-2905 

1-0712647 

1-0786616 

1-0864946 

59 

2 

1-0.")16612 

1-0978323 

1-0644033 

1-0713814 

1-0787885 

1-0866289 

58 

3 

1-0517608 

1-0579390 

1-0645163 

1-0715043 

1-0789156 

1-0867631 

57 

| 

1-0518606 

1-0530453 

1-0646294 

1-0716244 

1-0790427 

1-0868979 

5G 

5 

1-0519605 

1-0581517 

1-0647425 

1-0717445 

1-0791700 

1-0870326 

55 

6 

1-0520604 

1-0582583 

1-0648558 

1-0718647 

1-0792975 

1  -087  1675 

54 

7 

1-05-21605 

1-0583649 

1-0649693 

1-0719851 

1-0794250 

1-0873021 

53 

g 

1  -0522607 

1-0584717 

1-0(550828 

1-0721056 

1-0795527 

1-0874375 

52 

9 

1  '0523610 

1-0585786 

1-0651964 

1-0722262 

1-0796805 

1-0875727 

51 

10 

1-0524614 

1-0586855 

1-0653102 

1-0723469 

1-0798084 

1-0877080 

50 

H 

1-0525619 

1-0587925 

1-0654240 

1-0724678 

1-0799364 

1-0878435 

49 

l-052(i625 

1-0588999 

1-0655380 

1-0725887 

1-0800646 

1-0879791 

4S 

3 

1-0527633 

1-0590072 

1-0656521 

1-0727098 

1-OS01928 

1-0881148 

47 

1 

1-05-28641 

1-0591146 

1-0657663 

1-0728310 

1-0803212 

1-0882506- 

46 

15 

1-0529651 

1-0592221 

1-0638S07 

1-0729523 

1-0804497 

1-0883866 

45 

16 

1-0530(561 

1-0593298 

1-0659951 

1-0730737 

1  C805784 

1-088522(5 

44 

17 

1-0531673 

1-0594376 

1-0661097 

1-0731953 

1-OS07071 

1  0886589 

43 

18 

1-0532685 

1-0595451 

1-0662243 

1-0733170 

-10808360 

1  0887952 

4-2 

19 

1-0533699 

1-0596531 

1-0663391 

1-0734388 

1-0809650 

1-08S9317 

41 

20 

1-0534714 

1-0597615 

1-0664540 

1-0735607 

1-0810942 

10890682 

40 

21 

1-0535730 

1-0598697 

1-0665690 

1-073(5827 

1-0812234 

1-0892050 

39 

22 

1-0536747 

1-0599781 

1-0(566843 

1-0738048 

1-0813528 

1-0893418 

38 

23 

1-0537765 

l-06008f)5 

1-0667994 

1-0739271 

1-0814823 

1-0894788 

37 

21 

1-0538785 

1-0601951 

1-0669148 

1-0740495 

1-0816119 

1-08S6159 

3« 

25 

1-0539805 

1  -0603037 

1-0670302 

1-0741720 

1-0817417 

1-0897531 

35 

26 

1-0540826 

1-0604125 

1-0671458 

1-0742946 

1-0818715 

1-0898904 

34 

27 

1-0541849 

1-0605211 

1-0672615 

1-0744173 

1-0820015 

1-0900279 

33 

28 

1-0542873 

1-0606301 

1-0673774 

1-0745402 

1-08-21316 

1-0901655 

32 

29 

1-0543897 

1-0607395 

1-0674933 

1-0746631 

1-0822618 

1-0903032 

31 

30 

1-0544923 

1-0608487 

1-0676094 

1-0747862 

1-0823922 

1-0904411 

30 

31 

1-0545950 

1-0609580 

1-0677255 

1-0749095 

1-0825227 

1-0905791 

29 

32 

1-054G978 

1-0610675 

1-0678418 

1-0750328 

1-0826533 

1-0907172 

28 

33 

1-0548007 

1-0611770 

1-0679582 

1-0751562 

1-0827840 

1-0908554 

27 

34 

1-0549037 

1-0612867 

1-0680747 

1-0752798 

'1-0829149 

1-0909938 

96 

35 

1-0550068 

1-0613965 

1-OU81914 

1-0754035 

1-0830458 

1-0911323 

25 

36 

1-0551101 

1-0615064 

1-0683081 

1-0755273 

1-0831769 

1-0912709 

21 

37 

1-0552134 

1-0616161 

1-0684250 

1-0756512 

1-0833081 

1-0914097 

23 

38 

1-0553169 

1-0617265 

1-0685420 

1-0757753 

1-0834395 

1-0315485 

22 

39 

1-0554204 

1-0618367 

1-0686591 

1-0758995 

1-0835709 

1  -0916876 

21 

10 

1-0555211 

1-0619171 

1-0687763 

1-0760237 

1-0837025 

1-0918267 

20 

41 

1-0556279 

1-0620575 

1-0688936 

1-0761481 

1-0838342 

1-0919659 

19 

42 

1-0557318 

1-0621681 

1-0690110 

1-0762727 

1-0839661 

1-0921053 

18 

43 

1-0558358 

1-06-22788 

1-0691286 

1-0763973 

1-0840980 

1-0922448 

17 

41 

1-0559399 

1-0623896 

1-0692463 

1-0765221 

1-0842301 

1-0923845 

16 

15 

1-0560441 

1-0625005 

1-0693641 

1-0766470 

1-0843623 

1-0925243 

15 

46 

1-05(51485 

1-0626115 

1-0694820 

1-0767720 

1-0844947 

1-0926642 

14 

47 

1-0562529. 

1-06-27227 

1-0096000 

1-0768971 

1-0846271 

1-0928042 

13 

48 

1-0563575 

1-0628339 

1-0697182 

1-0770224 

1-0847597 

1-0929444 

12 

49 

1-0564621 

1-0629453 

1-0698364 

1-0771477  , 

1-0848924 

1-0930846 

11 

50 

1-0565669 

1-0630568 

1-0699548 

1-0772732 

1-0850252 

1-0932251 

10 

51 

1-0566718 

1-0631681 

1-0700733 

1-0773988 

1-0851582 

1-0933656 

9 

52 

1-0567768 

1-063-2801 

1-0701919 

1-0775246 

1-0852913 

1-0935003 

8 

53 

1-0568819 

1-0633919 

1-0703103 

1-0776504 

1-0854245 

1-0936471 

7 

51 

1-0569871 

1-0635038 

1-0704295 

1-0777764 

1-0855578 

1-0937880 

6 

55 

1-0570924 

1-06.36158 

1-0705484 

1-0779025 

1-0856912 

1-0939291 

5 

56 

1-0571978 

1-0637280 

1-0706675 

1-0780287 

1-0858248 

1-0940702 

4 

57 

1-0573034 

1-06384J3 

1-0707867 

1-0781550 

1-0859585 

1-0942116 

3 

01 

1-0574090 

1-0639527 

1-0709060 

1-0782815 

1-0800924 

1-0943530 

2 

59 

1-0575148 

1-0640652 

1-0710254 

1-0784080 

1-0862263 

1-0944946 

60 

1-0576207 

1-0641778 

1-0711450 

1-0785347 

1-0863604 

1-0946363 

0 

71° 

70° 

69° 

68° 

67° 

66° 

/ 

COSECANTS. 

29; 


TABLE  XI.— SECANTS  AND  COSECANTS. 


SECANTS. 

/ 

24° 

25° 

26° 

27° 

28° 

29° 

t 

0 

1-0946363 

1-1033779 

1-1126019 

1-1223262 

1-1325701 

1-1433541 

60 

1 

1-0947781 

1-1035277 

1-1127599 

1-1224927 

1-13-27453 

1-1435385 

59 

1-0949201 

1-1036775 

1-1129179 

1-1226592 

1-1  329207 

1-1437231 

58 

3 

1  -09506-2-2 

1-1038275 

1-1130701 

1-1228259 

1-1330962 

1-1139078 

57 

4 

1  -0952044 

1-1039777 

1-1132345 

1-1229928 

1-1332719 

1-1440927 

56 

5 

1-0953107 

1-1041279 

1-1133929 

1-1231598 

1-1334478 

1-1442778 

55 

6 

1-0954892 

1-1042783 

1-1135516 

1-1233269 

1-1336238 

1-1444630 

54 

7 

1-0956318 

1-1044289 

1*1137103 

1-1231942 

1-1337999 

1-1446484 

53 

8 

1-0957746 

1-1045795 

l-1138(i92 

1-1236016 

1-1339762 

1-1448339 

52 

9 

1-0959174 

1-1047303 

1-1140282 

M23S292 

1-1341527 

1-1450196 

51 

10 

1-0960604 

1-1048813 

1-1141874 

1-1239969 

1-1313293 

1-1452055 

50 

11 

1-0962036 

1-1050324 

1-1143467 

1-1211618 

1-1345060 

1-1453915 

49 

12 

1-0963468 

1-1051836 

1-11450G-2 

1-1243328 

1-1340829 

1-1455776 

48 

13 

1-0964902 

1-1053319 

MH6053 

1-1245010 

1-1348500 

1-1457639 

47 

14 

1-0906337 

1-1054864 

1-1148255 

1-1246693 

1  1350372 

1-1459504 

46 

15 

1-0967774 

1-1056380 

1-11*9854 

1-1248377 

1-135-2146 

1-1461371 

45 

16 

1-0969212 

1-1057898 

1-1151454 

M250063 

1-1353921 

1-1463238 

44 

17 

1-0970651 

1-1059417 

1-1153056 

1-1251750 

1-1355697 

1-1465108 

43 

18 

1-0972091 

1-1060937 

1-1154659 

1-1253439 

1-1357476 

1-1466979 

42 

19 

1-0973533 

1-1062458 

l-1156l'G3 

1-1255130 

1-1359255 

1-1468852 

41 

20 

1-0974976 

1-1063981 

1-1157869 

1-1256821 

1-1361036 

1-1470726 

40 

21 

1-0976420 

1-106550G 

1-1159476 

1-1258514 

1-1362819 

1-1472602 

39 

22 

1-0977866 

1-1067031 

1-1161084 

M  960209 

1-1364603 

1-1474479 

38 

23 

1-0979313 

M068558 

1-1162691 

l-126iy05 

1  -1300389 

1-1476358 

37 

21 

1-0980761 

1-1070087 

1-1164300 

1-1203003 

M368176 

1-1478-239 

36 

25 

1-0982211 

M071616 

1-1165919 

1-1265302 

1-1369965 

1-1480121 

35 

26 

1-0983662 

1-1073147 

1-1167533 

1-1267003 

1-1371755 

1-1482005 

34 

27 

1-0985114 

1-1074C80 

1-1169118 

1-1208705 

1-1373547 

1-1483890 

33 

28 

1-0986568 

1-1076214 

1-1  170706 

1-1270408 

1-1375341 

1-1185777 

32 

29 

1-0988023 

1-1077749 

1-1172384 

1-1272113 

1-1377135 

1-1487065 

31 

80 

1-0989479 

1-1079285 

1-1174004 

1-1273819 

1-1378932 

1-1489555 

30 

31 

1-0990336 

1-1080823 

1-1175625 

1-1275527 

J  -1380730 

1-1491447 

29 

32 

1-0992395 

1-1082363 

1-1177248 

1-1277237 

1-1382529 

1-1493340 

23 

33 

1-0993855 

1-1083903 

1-1178872 

1-1278948 

1-1381330 

1-1495235 

27 

31 

1-0995317 

1-1085445 

1-1180498 

l-1280CfiO 

1-1380133 

1-1497132 

26 

35 

1-0996779 

1-1086989 

1-1182124 

1-1282374 

1-1387937 

1-1499030 

25 

36 

1-0998243 

1-1088533 

1-1183753 

1-1284089 

1-1389742 

1-1500930 

24 

37 

1-0999709 

1-1090079 

M1853o3 

1-1285806 

1-1391550 

1-1502831 

23 

38 

1-1001175 

1-1091627 

1-1187014 

1-1287524 

1-1393358 

1-1501734 

23 

39 

1-1002644 

1-1093176 

1-1188647 

1-1289244 

1-1390169 

1-1500638 

21 

iu 

1-1004113 

1-10S4726 

M190281 

1-1290965 

1-13U6980 

1-1508544 

20 

41 

M005584 

1-1096277 

1-1191916 

1-1292687 

1-1398794 

1-1510452 

19 

42 

1-1007056 

1-1097830 

1-1193553 

1-1294412 

1-1400608 

1-1512361 

18 

43 

1-1008529 

1-1099385 

1-1195191 

1-1206137 

1-1102425 

1-1514272 

17 

44 

1-1010004 

1-1100940 

1-119G831 

1-1297864 

1-UU42J3 

1-1510185 

16 

45 

1-1011480 

1-1102498 

1-1198472 

1-1299593 

1-1400062 

1-1518099 

15 

46 

1-1012957 

1-1104056 

1-1200115 

1-1301323 

1-1407883 

1-1520015 

14 

47 

1-1014436 

1-1105616 

i-1201739 

1-1303035 

1-1409706 

1-1521932 

13 

48 

1-1015916 

1-1107177 

1-1203405 

1-1304788 

1-1411530 

1-1523801 

12 

49 

1-1017397 

1-1108740 

1-1:4)5051 

M30G522 

1-1413356 

1-1525772 

11 

50 

1-1018879 

M1103J4 

1-1200700 

1-1308258 

1-1415183 

1-1527094 

10 

51 

1-1020363 

1-1111869 

1-1208350 

M309996 

1-1417012 

1-1529618 

9 

52 

1-1021849 

1-1113436 

1-1210001 

1-1311735 

1-1418842 

1-1531513 

8 

53 

1-1023335 

1-1115004 

1-1211S53 

1-1313475 

1-1,20674 

1-1533170 

7 

54 

1-1024823 

1-1116573 

1-1213308 

M315217 

1-1422507 

1-1535399 

6 

55 

1-1026313 

1-1118144 

1-12149C3 

1-1  3  16961 

1-1424342 

1-1537329 

0 

56 

1-1027803 

1-1119716 

1-121GC20 

1-1318706 

1-1426179 

1-1539261 

4 

67 

1-1029295 

1-1121290 

1-1218278 

1-1320452 

1-1428017 

1-1541195 

3 

58 

1-1030789 

1-1122865 

1  -1219938 

1-1322200 

1-1429857 

1-1543130 

2 

59 

1-1032283 

1-1124442 

1-1221600 

1-1323950 

1-1431698 

M545007 

1 

60 

M033779 

1-1120019 

M223282 

1-1325701 

1-1433541 

1-1547005 

0 

t 

65° 

64° 

63° 

62° 

61° 

60° 

/ 

COSECANTS. 

TABLE  XL-SECANTS  AND  COSECANTS. 


.  SECANTS. 

/ 

30° 

31° 

32° 

33° 

34° 

35° 

i 

0 

11547005 

1-1666334 

1-1791784 

1-1923633 

1-2062179 

1-2207746 

60 

1 

1-1518945 

1-1668374 

1-1793928 

1-1925886 

1-2064547 

1-2210233 

59 

2 

1-1550887 

1-1670416 

1-1796074 

1-1928142 

1-2006917 

1-2212723 

58 

3 

1-1552830 

1-1672459 

1-1798222 

1-1930399 

1-2069288 

1-2215215 

57 

4 

1-1554775 

1-1674504 

1-1800372 

1-1932658 

1-2071662 

1-2217708 

56 

5 

1-1556722 

1-1676551 

1-1802523 

1-1934918 

1-2074037 

1-2220204 

55 

6 

1-1558670 

1-1678599 

1-1804676 

1-1937181 

1-2076415 

1-2222702 

54 

7 

1-1560620 

1-1680649 

1-1806831 

1-1939446 

1-2078794 

1-2225202 

53 

8 

M56257S 

1-1682701 

1-1808988 

1-1941712 

1-2081175 

1-2227703 

52 

9 

1-1564525 

1-1684755 

1-1811146 

1-1943980 

1-2083559 

1-2230207 

51 

10 

1-1560480 

1-1686810 

1-1813307 

1-1946251 

1-20^5944 

1-2232713 

50 

11 

1-1568436 

1-1688867 

1-1815469 

1-1948523 

1-2088331 

1-2235222 

49 

12 

1-1570394 

1-1690926 

1-1817633 

1-1950796 

1-2090720 

1-2237732 

48 

13 

1-1572354 

1-1692986 

1-1819798 

1-1953072 

1-2093112 

1-2240244 

47 

11 

1-1574315 

1-1695048 

1-1821966 

1-1955350 

1-2095505 

1-2242758 

46 

15 

1-1576278 

1-1697112 

1-1824135 

1-1957629 

1-2097900 

1-2245274 

45 

16 

1-1578243 

1-1699178 

1-1826306 

1-1959911 

1-2100297 

1-2247793 

44 

17 

1-1580209 

1-1701245 

1-1828479 

1-1962194 

1-2102696 

1-2250313 

43 

18. 

1-1582177 

1-1703314 

1-1830654 

1-1964479 

1-2105097 

1-2252836 

42 

19 

1-1584146 

1  '1705385 

T1832830 

1-1966767 

1-2107500 

1-2255361 

41 

20 

1  -1586118 

1-1707457 

1-1835008 

1-1969056 

1-2109905 

1-2257887 

40 

21 

1-1588091 

1-1709531 

1-1837188 

1-1971346 

1-2112312 

1-2260416 

39 

22 

1-1590065 

1-1711607 

1-1839370 

1-1973639 

1-2114721 

1-2262947 

38 

23 

1-1592041 

1-1713685 

1-1841554 

1-1975934 

1-2117132 

1-2265180 

37 

24 

1-1591019 

1-1715764 

1-1843739 

1-1978230 

1-2119545 

1-2268015 

3d 

25 

1-1595999 

1-1717845 

1-1845927 

1-1980529 

1-2121960 

1-2270552 

35 

26 

1-1597980 

1-1719928 

1-1848116 

1-1982829 

1-2124377 

1-2273091 

34 

27 

1-1599963 

1-1722013 

1-1850307 

1-1985131 

1-2126795 

1-2275633 

33 

23 

1-1601947 

1-1724099 

1-1852500 

1-1987435 

1-2129216 

1-2278176 

32 

29 

1-1603U33 

1-1726187 

1  -1854694 

1-1989741 

1-2131639 

1-2280722 

31 

30 

1-1605921 

1-1728277 

1-1856890 

1-1992049 

1-21340G4 

1-2283269 

30 

31 

1-1607911 

•1730368 

1-1859089 

1-1994359 

1-2136191 

1-2285819 

29 

32 

1-1609902 

•1732462 

1-1861289 

1-199W71 

1-2138920 

1-2288371 

28 

33 

1-1611894 

•1734557 

1-1863490 

1-19981)85 

1-2141351 

1-2290924 

27 

31 

M6138S9 

•1736-053 

1-1863694 

1-2001300 

'1-2143784 

1-2293480 

26 

35 

1-1615885 

•1738752 

1-1867900 

1-2003618 

1-2146218 

1-2296039 

25 

36 

1-1617883 

•1740852 

1-1870107 

1-2005937 

1-2148655 

1-2298599 

24 

37 

M619882 

•1742954 

1-1872310 

1-2008258 

1-2151094 

1-2301161 

23 

38 

1-1621883 

•1745058 

1-1874527 

1-2010582 

1-2153535 

1-2303725 

22 

39 

1-1623886 

•1747163 

1-1876740 

1-2012907 

1-2155978 

1  -2306292 

21 

40 

1-1625891 

•1749270 

1-1878954 

1-2015234 

T2158423 

1-2308861 

20 

41 

1-1627897 

•1751379 

1-1881171 

1-2017563 

1-2160870 

1-2311432 

19 

42 

1-1G29905 

•1703490 

1-1883389 

1-2019894 

1-2163319 

1-2314004 

18 

43 

1-1631914 

•1705603 

1-1685609 

1-2022226 

1-2165770 

1-2316579 

17 

41 

1-1633925 

•1757717 

1-1867831 

1-2024561 

1-2168223 

1-2319156 

16 

45 

1-1635938 

•1759833 

1-1890055 

1  -2026898 

1-2170673 

1-2321736 

15 

46 

1-1637953 

•17C1951 

1-1892280 

1-2029236 

1-2173135 

1-2324317 

14 

47 

1-1639969 

•1764070 

1-1894508 

1-2031577 

1-2175594 

1  •2320!X)0 

13 

48 

1-1641987 

•1766191 

1-1896737 

1-2033919 

1-2178055 

1  '2329486 

12 

49 
50 

1-1644007 
1-1646028 

•1768314 
•1770439 

1-1898963 
1-1901201 

1  -2036261  , 
1-2038610 

1-2180518 
1-2182983 

1-2332(374 
1-2334G61 

1) 
10 

51 

1-1648051 

1-1772566 

1-190343G 

1-2040958 

1-2185450 

1-2337256 

9 

52 

1-1650076 

1-1774694 

1-1D05K73 

1-2043308 

1-2187919 

1*2339850 

8 

53 

1-1652102 

1-1776824 

1-11)07911 

1-2045660 

1-2190390 

1-2342446 

7 

51 

1-11554130 

1-1778956 

1-1910152 

1-2048014 

1-2192864 

T2345044 

6 

55 

1-1656160 

1  -178.  1069 

1-1912394 

1-2050370 

1-2195339 

1-2347645 

5 

56 

1-1658191 

1-1783225 

1-1914638 

1-2052728 

1-2197816 

1-2350248 

4 

57 

1-1660224 

1-17&5362 

•M91K8S4 

1-2055088 

1-2200296 

1  '2352852 

3 

58 

1-1C&-2239 

M787001 

1-191SH32 

1-2057450 

1-2202777 

1  -2355453 

2 

59 

1-1664296 

1-17S9U42 

1-19213-J1 

1-2059814 

1-2200260 

1  -235S069 

1 

CO 

1-1666331 

1-1791784 

1-19236J3 

1-2062179 

1-2207746 

1-2360680 

0 

• 

69° 

68° 

67° 

66° 

65° 

64° 

/ 

COSECANTS. 

299 


TABLE   XL— SECANTS  AND  COSECANTS. 


SECANTS. 

36° 

37° 

38° 

39° 

40° 

41° 

9 

1-2360680 

1-2521357 

1-2690182 

1-2S67596 

1-3051073 

1-3250130 

60 

1-2363293 

1-2521102 

1-2693067 

1-2370628 

1-3057261 

1-3253182 

59 

1-2365909 

1-2526850 

1-2695955 

1-2873663 

1-3060451 

1-3256837 

53 

1-2368526 

1-2529G01 

1-2698845 

1-2876700 

1-3063614 

l-3iU0191 

57 

1-2371146 

1-2532353 

1-2701737 

1-2879740 

1-3066839 

1-3203554 

56 

1-2373768 

1-2535108 

1-2704632 

1-2882782 

1-3070038 

1-326CJ918 

55 

1-2376393 

1-2537865 

1-2707529 

1-2885827 

1-3073239 

1  -3270281 

04 

1-2379019 

1-2510625 

1-2710429 

1-2888875 

1-3076142 

1-3273653 

53 

1-2381647 

1-2513387 

1-2713331 

1-2891925 

1-3079619 

1-3277024 

52 

1-2384278 

1-2516151 

1-2716235 

1-2894977 

1-3082853 

1-3280399 

51 

1-2386911 

1-2518917 

1-2719142 

1-2898032 

1-3086069  . 

1-3283776 

CO 

10 

1-2389546 

1-2551685 

1-2722052 

1-2901090 

1-3089284 

1-3287156 

49 

11 

1-2392183 

1-2554456 

1-2721963 

1-2904150 

1-3092501 

1-3290539 

43 

12 

1-2394823 

l-25572-:3 

1-2727877 

1-2907213 

1-3095720 

1-3293925 

47 

13 

1-2397464 

1-2060005 

1-2730794 

1-2910278 

1-3098943 

r32'J7314 

46 

14 

1-2400108 

1-2562782 

1-2733712 

1-2913316 

1-3102168 

1-3300706 

45 

15 

1  2102754 

1-2565502 

1-2736634 

1-2916116 

1-3105396 

1-3301100 

44 

16 

1-2405402 

1-25G8315 

1-2739557 

1-2919189 

1-3108676 

1-3307197 

43 

17 

1-2108052 

1  '257  11  29 

1-2712184 

1-2922564 

1-3111860 

1-3310897 

42 

18 

1-2410704 

1-2573916 

1-2745412 

1-2925642 

1-3115095 

1-3314301 

41 

19 

1-2413339 

1-2576705 

1-2748313 

1-2928723 

1-3118334 

1-3317707 

40 

20 

1-211G016 

1-2079497 

1-2751276 

1-2931806 

1-3121575 

1-3321115 

39 

21 

1-2118675 

1  -'2582291 

1-2751212 

1-2934892 

1-3121820 

1-3321527 

38 

22 

1-2121336 

1-2585087 

1-2757151 

1-2937980 

1-3128066 

1-33-27912 

37 

23 

1-2423999 

1-2587885 

1-2700091 

1-2911071 

1-3131316 

1-3331359 

3<S 

24 

1-2426665 

1  -2590686 

1-2763034 

1-2914164 

1-3134563 

1-3331779 

35 

25 

1-2429333 

1-2593489 

1-2765980 

1-2917260 

1-3137823 

1-3338203 

31 

26 

1-2432003 

1-2096294 

1  -276^28 

1-2950359 

1-3111081 

1-3311029 

33 

27 

1-2134675 

1-2099102 

1-2771878 

1-2953160 

1-3114341 

1-3315058 

32 

23 

1-2437349 

1-2601912 

1-2774831 

1-2956564 

1-3147604 

1-3348189 

31 

n 

1-2140026 

1-2604724 

1-2777787 

1-2959070 

1-3150870 

1-3301924 

30 

30 

1-2442704 

1-2607539 

1-2780744 

1-2962779 

1-3154139 

1-3355362 

29 

31 

1-2415385 

1-2610356 

1-2783705 

1-2960890 

1-3157410 

1-3358802 

28 

32 

1-2118069 

1-2613175 

1-278G667 

1-2909004 

1-3160684 

1-3362-216 

27 

33 

1-2450754 

1-2615997 

1-2789632 

1-2972121 

1-3163961 

1-3360092 

20 

84 

1-2153142 

1-2618820 

1-2792600 

1-2975240 

1-3167240 

1-3369141 

25 

85 

1-2456131 

1-2621617 

1-2790570 

1-2978362 

1-3170523 

1-3372594 

21 

36 

1-24588-23 

1-2621175 

1  -2798543 

1-29814S7 

1-3173808 

1-8376019 

23 

37 

1-2461518 

1-26-27306 

1-2801518 

1-2981614 

1-3177096 

1-3379507 

22 

38 

1-2161214 

1-2630  HO 

1-2801195 

1-2987743 

1-3180386 

1-8.182908 

21 

39 

1-2166913 

1-2G32975 

1-2807475 

1-2990876 

1-3183680 

1-3386132 

28 

40 

1-2469614 

1-2635813 

1-281Q157 

1-5991011 

1-3186976 

1-33S9898 

19 

41 

1-2472317 

l-263b(J53 

1-2813142 

1-21/97148 

1-3190274 

1-3393368 

18 

42 

1-2475022 

1  -'2611198 

1-2816130 

1-3000288 

1-3193576 

1-3390841 

17 

43 

1-2477730 

1-2611311 

1-2819119 

1-30C3131 

1-3196881 

1-3400316 

16 

44 

1-2180410 

1-2047188 

1-2822412 

1-3006576 

1-3200188 

1-3103795 

15 

45 

1-2183152 

1-2650038 

1-2525407 

1-3009724 

1-3203498 

1-3407276 

14 

46 

1-2485866 

1-2652890 

1-2828401 

1-3012875 

1-3200810 

1-3410761 

13 

47 

1-2488583 

1-2655745 

1-2831404 

1-3010028 

1-3210126 

1-3414248 

12 

48 

1-2491302 

1-2G08G01 

l-2»3410f? 

1-3019184 

1-3213144 

1-3417738 

11 

49 

1-24940~>3 

1-2661460 

1-2837411 

1-302-2313 

1-3216705 

1-3121232 

10 

00 

1-2196746 

1-2664322 

1-2840418 

1-3025004 

1-3220089 

1-3121728 

9 

01 

1-2499471 

1-26G7186 

1-2843423 

1-3028667 

1-3223116 

1-3128227 

8 

02 

1-2502199 

1-2070052 

1-2346140 

1-3031831 

1-3220745 

1  3131729 

7 

53 

1-2501929 

1-2672S21 

1-2849455 

1-3035003 

1-3230078 

1-3435234 

6 

04 

1-2507661 

1-2675792 

1-2852472 

1-3038175 

1-3233413 

1-3133712 

0 

05 

1-2510396 

1-2678665 

1-2855492 

1-3041349 

1-3236750 

1-341225$ 

4 

06 

1-2513133 

1-2681041 

1-2858514 

1-3044526 

1-3210091 

1-3440767 

3 

07 

1-2516872 

1-2681419 

1-2861539 

1-3047706 

1  •3243135 

1-3449284 

2 

53 

1-2518613 

1-2687299 

1-2864566 

1-3050888 

1-324G781 

1-3102S04 

1 

09 

1-2521357 

1-2690182 

1-2867596 

1-3054073 

1-8250120 

1-3106327 

0 

60 

63° 

52° 

61° 

60° 

49° 

48° 

t 

COSECANTS. 


AINU    UUSJ1.UAJN  1  R>. 


DECANTS. 

/ 

42° 

43° 

44° 

45° 

46° 

47- 

/ 

0 

1  -345(5327 

1-3673275 

1-3901636 

1-4142136 

1-4395565 

1-4662792 

60 

1 

1-3459853 

1-3676985 

1-390J543 

1-4146251 

1-4399904 

1-4667368 

59 

2 

1-3463382 

1-3680699 

1-3909453 

1-4150370 

1-4404246 

1-4671943 

58 

3 

1-34W5914 

1-3684416 

1-3913366 

1-4154493 

1-4408592 

1-4676532 

57 

4 

1-3470449 

1-3688136 

1  -3917283 

1-4158619 

1-4412941 

1-4681120 

56 

5 

1-3473987 

1-3691859 

1-3921203 

1-4162749 

1-4417295 

1-4G85713 

55 

6 

1-3477528 

1-3695586 

1-3925127 

1-4166883 

1-4421652 

1-4690309 

54 

7 

1-3481072 

1-3699315 

1-3929054 

1-4171020 

1-4426013 

1-4694910 

53 

8 

1-3484G19 

1-3703048 

1-3932985 

1-4175161 

1-4430379 

1-46995U 

52 

9 

1-3188168 

1-3706784 

1-3936918 

1-4179306 

1-4434748 

1-4704123 

51 

10 

1-3491721 

1-3710523 

1-3940856 

1-4183454 

1-4439120 

1-4708736 

50 

11 

1-3495277 

1-3714266 

1-3914796 

1-4187605 

1-4443497 

1-47M354 

49 

12 

1-3498836 

1-3718011 

1-3948740 

1-4191761 

1-4447878 

1-4717975 

48 

13 

1-3502398 

1-3721760 

1-3952688 

1-4195920 

1-4452262 

1-4722600 

47 

14 

1-3505963 

1-3725512 

1-3956639 

l-4->00082 

1-4*56651 

1-4727230 

46 

15 

1-3509531 

1-3729868 

1-3960593 

1-4204248 

1-4461043 

1-4731864 

45 

16 

1-3513102 

1-3733025 

1-3964551 

1-4208418 

1-4465439 

1-4736502 

44 

17 

1-3516677 

1-3736788 

1-3968512 

1-4212592 

1-4469839 

1-4741144 

43 

18 

1-3520254 

1-3740553 

K-3972477 

1-4216769 

1-4474243 

1-4745790 

42 

19 

1-3523834 

1-3744321 

1-3976445 

1-4220950 

1-4478651 

1-4750440 

41 

20 

1-3527417 

1-3748092 

1-3980416 

1-4225134. 

1-4483063 

1-4755095 

40 

21 

1-3531003 

1-3751867 

1-3984391 

1-4229323 

1-4487478 

1-4759751 

39 

22 

1-3531593 

1-3755645 

1-3988369 

1-4233514 

1-4491898 

1-4764417 

33 

23 

1-3538185 

1-3759426 

1-3992351 

1-4237710 

1-4496322 

1-4769084 

37 

24 

1-3541780 

1-3763210 

1-3996336 

1-4241909 

1-4500749 

1-4773755 

36 

25 

1-3545379 

1-3766998 

1-4000325 

1-4246112 

1-4505181 

1-4778431 

35 

26 

1-3548980 

1-3770789 

1-4004317 

1-4250319 

1-4509616 

1-4788111 

34 

27 

1-3552585 

1-3774583 

1-4008313 

1-4254529 

1-4514055 

1-4787795 

33 

28 

1-3556193 

1-3778380 

1-4012312 

1-425874S 

1-4518498 

1-4792483 

32 

-29 

1-3559803 

1-3782181 

1-4016315 

1-4262961 

1-4522946 

1-4797176 

31 

30 

1-3563417 

1-3785985 

1-4020321 

1-4267182 

1-4527397 

1-4801872 

30 

31 

1-3567034 

1-3789792 

1-4024330 

1-4271407 

1-4531852 

1-4806573 

29 

32 

1-3570654 

1-3793(502 

1-4028343 

1-4275636 

1-4536311 

1-4811278 

23 

33 

1-3574277 

1-3797416 

1-4032360 

1-4279868 

1-4540774 

1-4815988 

27 

34 

1-3577903 

1-3801233 

1-4036380 

1-4284105 

1-4545241 

1-4820702 

26 

35 

1-3581532 

1-3805053 

1-4040403 

1-4288345 

1-4549712 

1-4825420 

25 

36 

1-3585164 

1-3808877 

1-4044430 

1-4292588 

1-4554187 

1-4830142 

24 

37 

1-3588800 

1-3812704 

1-4048461 

1-4296836 

1-4558666 

1-4834868 

23 

38 

1-3592438 

1-3816534. 

1-4052494 

1-4301087 

1-4563149 

1-4839599 

22 

39 

1-3596080 

1-3820367 

1-4050532 

1-4305342 

1-4567636 

1-4844334 

21 

40 

1-3599725 

1-3824204 

1-4060573 

1-4309600 

1-4572127 

1-4849073 

20 

41 

1-3603372 

1-3828044 

1-4064(517 

1-4313863 

1-4576621 

1-4853817 

19 

42 

1-  3607023 

1-3831887 

1-4068665 

1-4318129 

1-4581120 

1-4858565 

18 

43 

1-3610677 

1-3835734 

1-4072717 

1-4322399 

1-4585623 

1-4863317 

17 

44 

1-3614334 

1-3839584 

1-4076772 

1-4326672 

1-4590130 

1-4868078 

16 

45 

1-3617995 

1-3843437 

1-4080831 

1-4330950 

1-4594641 

1-4872834 

15 

46 

1-3621658 

1-3847294 

1-4084893 

1-4335231 

1-4599156 

1-4877599 

14 

47 

1-3625324 

1-3851153 

1-4088958 

1-4339516 

1-4603675 

1-4882369 

13 

48 

1-3628994 

1-3855017 

1-4093028 

1-4343805 

1-4608198 

1-4887142 

12 

49 

1-3632667 

1-3858883 

1-4097100 

1-4348097 

-  1-4612726 

1-4891920 

11 

50 

1-3636343 

1-3862753 

1-4101177 

1-4352393 

1-4617257 

1-4896703 

10 

51 

1-3640022 

1-3866626 

1-4105257 

1-4356693 

1-4621792 

1-4901489 

9 

52 

1-3643704 

1-3870503 

1-4109340 

1-4360997 

1-4626331 

1-4906280 

8 

53 

1-3647389 

1-3874383 

1-4113427 

1-4365305 

1-4630875 

1-4911076 

7 

54 

1-3651078 

1-3878286 

1-4117517 

1-4  3696  16 

1-4635422 

1-4915876 

6 

55 

1-3654770 

1-3882153 

1-4121612 

1-4373932 

1-4639973 

1-4920680 

5 

56 

1-3658464 

.  1-3886043 

1-4125709 

1-4378251 

1-4644529 

1-4925483 

4 

57 

1-3662162 

1-3889936 

1-4129810 

1-4382574 

1-4649089 

1-4930301 

3 

58 

1-3665863 

1  -3893832 

1-4133915 

1-4386900 

1-4653652 

1-4935118 

2 

59 

1-3669567 

1-3897733 

1-4138024 

1-4391231 

1-4658220 

1-4939940 

1 

60 

1-3673275 

1-3901636 

1-4142136 

1-4395565 

1-4662792 

1-4944765 

0 

/ 

47° 

46° 

45° 

44° 

43° 

42° 

/ 

COSECANTS. 

J 

SOI 


TABLE  XI.— SECANTS  AND   COSECANTS. 


SECANTS. 

/ 

48° 

49° 

50° 

51° 

52° 

53° 

0 

1-4911765 

1-5242531 

1-5557238 

1-5890157 

1-6242692 

1-6616101 

60 

1 

1-4949596 

1-5247634 

1-5562634 

1-5895868 

1-6248743 

1-6622819 

2 

1-4954431 

1  -5252741 

1-5568035 

1-5901584 

1-6254799 

1-6629213 

j 

3 

1-4959270 

T5257854 

1-5573441 

1-5907306 

1-6260861 

1-6635673 

57 

4 

1-19(54113 

1-5262971 

1-5578852 

1-5913033 

1-6266929 

1-6642110 

56 

5 

1-4908961 

1-5268093 

1-5584268 

1-5918766 

1-6273003 

1-6648553 

55 

6 

1-4973813 

1-5273219 

1-5589689 

1-5924504 

1-6279083 

1-6655002 

54 

7 

1-4978670 

1-5278351 

1-5595115 

1-5930247 

1-6285169 

1-6661458 

53 

8 

1  -1983531 

1-5283487 

1-5600546 

1-5935996 

1-6291261 

1-6667920 

52 

9 

1-4988897 

1-5288627 

1-5605982 

1-5941751 

1-6297359 

1-6674389 

51 

10 

1-4993267 

1-5293773 

1-5611424 

1-5947511 

1-6303462 

1-6680864 

50 

11 

1-4998141 

1-5298923 

1-5616871 

1-5953276 

1-6309572 

16687345 

49 

12 

1-0003020 

1-5304078 

1-5622322 

1-5959043 

1-6315688 

1-6693833 

48 

13 

1-5007903 

1-530923S 

1-5627779 

1-5964821 

l'-6321809 

1-6700328 

47 

u 

1-5012791 

1-5314403 

1-5633241 

1-5970606 

1-6327937 

1-6706828 

46 

15 

1-5017C83 

1-5319572 

1-5638708 

1-5976394 

1-6334070 

1-6713336 

45 

16 

1-50  -22580 

1-5324746 

1-5644181 

1-5982187 

1-6340210 

1-6719850 

44 

17 

1  '5027481 

1-5329925 

1-5649658 

1-5987986 

1-6346355 

1-6726370 

43 

18 

1-5032387 

1-5335109 

1-5655141 

1-5993790 

1-6352507 

1-6732897 

a 

19 

1-5037297 

1-5340297 

1-5660628 

1-5999600 

1-6358661 

1-6739430 

41 

20 

1-5012211 

1-5345191 

1-5666121 

1-6005416 

1-6364828 

1-6745970 

40 

21 

1-5017131 

1-5350689 

1-5671619 

1-6011237 

1-6370997 

1-6752517 

39 

22 

1-5052054 

1-5355892 

1-5677123 

1-6017064 

1-6377173 

1-6759070 

38 

23 

1-5056982 

1-5361100 

1-5682631 

1-6022896 

1-6383355 

1-6765629 

37 

24 

1-5061915 

1-5366313 

1-5688145 

1-6028734 

1-6389512 

1-6772195 

36 

25 

1-5066S52 

1-5371530 

1-5693664 

1-6034577 

1-6395736 

1-6778768 

35 

26 

1T.071793 

1-5376752 

1-5699188 

1-6040126 

1-6401936 

1-67853  17 

31 

27 

1-5070739 

1-5381980 

1-5704717 

1-6046281 

1-6408142 

1-6791933 

33 

28 

1-50S1G90 

1-5387212 

1-5710252 

1-6052142 

1-6414354 

1-6798525 

32 

29 

1-5086S45 

1-5392449 

1-5715792 

1-6058008 

1-6420572 

1-6805124 

31 

30 

1-5091505 

1-5392690 

1-5721337 

1-6063879 

1-6126796 

1-6811730 

30 

31 

I-50965G9 

1-5402937 

1-5726887 

1-6069757 

1-6133027 

1-6818342 

29 

32 

1-510153S 

1-5408189 

1-5732443 

1-6075640 

1-6139263 

1-6824961 

28 

33 

1-5106511 

1-5413445 

1-5738004 

1-6081528 

1-6445506 

1-6831586 

27 

31 

1-5111489 

1-5418706 

1-5743570 

1-6087423 

1-6151754 

1-6838219 

26 

35 

1-5116472 

1-5423973 

1-5749141 

1-6093323 

1-6458009 

1-6844857 

25 

36 

1-5121459 

1-5429244 

1-5754718 

1-6099228 

1-6461270 

1-6851503 

24 

37 

1-5126150 

1-5134520 

1-5760300 

1-6105140 

1-6170537 

1-6858155 

23 

38 

1-5131446 

1-5439801 

1-5765887 

1-6111057 

1-6476811 

1-6864814 

22 

39 

1-5136417 

1-5445087 

1-5771479 

1-6116980 

1-6483090 

1-6S71479 

21 

40 

1-5141452 

1-5150378 

1-5777077 

1-6122908 

1-6489376 

1-6878151 

20 

41 

1-514(5462 

1-5155673 

1-5782680 

1-6128843 

1*6495668 

1-6884830 

19 

42 

1-5151477 

1-5160974 

1-5783289 

16134788 

1-6501966 

1-6891516 

18 

43 

1-5106496 

1-5466280 

1-5793902 

1-6140728 

1-6508270 

1-6898208 

17 

44 

1-5161520 

1-5471590 

1-5799521 

1-6146680 

1-6514581 

1-6904907 

16 

45 

1-5166548 

1-5176906 

1-5305146 

1-6152637 

1-6520898 

1-6911613 

15 

46 

1-5171581 

1-5482226 

1-5810776 

1-6158600 

1-6527221 

1-6918325 

14 

47 

1-5176619 

1-5487552 

1-5816411 

1-6164569 

1-6533550 

1-6925045 

13 

.48 

1-5181661 

1-5492882 

1-58-22051 

1-6170514 

1-6539885 

1-6931771 

12 

49 

1-5186708 

1-5498-218 

1-5827097 

1-6  176524 

1-6546227 

1-6938504 

11 

50 

1-5191759 

1-5503558 

1-5333348 

1-6182510 

1-6552575 

1-6945244 

10 

01 

1-5196815 

1-5508904 

1-5839005 

1-6188502 

1-6558929 

1-6951990 

9 

52 

1-5201876 

1-5514251 

1-5844667 

1-6194500 

1-6565290 

1-6958714 

8 

53 

l-5i'06942 

1-5519610 

1-5850334 

1-6200504 

1-6571657 

1-6965504 

7 

51 

1-5212012 

1-5524970 

1-5356007 

1-6206513 

1-6578030 

1-6972271 

6 

55 

1-5217087 

1-5530335 

1-5861.685 

1-6212523 

1-6584109 

1-6979044 

5 

5G 

1-5222166 

1-5535706 

1  -5S67369 

1-C218549 

1-6590795 

1-6985825 

4 

67 

1-5227-250 

1-5541081 

1-5S73058 

1-6224576 

1-6597187 

1-6992612 

3 

53 

1-52:12339 

1-5546462 

1-5878752 

1-6230609 

1-6603586 

16999407 

2 

59 

1-5237433 

1-5551848 

1-5884452 

1-0236648 

1-6609990 

1-7006208 

1 

€0 

1-5242D31 

1-5557238 

1-5890157 

1-6212692 

1-6616101 

1-7013016 

0 

/ 

41° 

40° 

39° 

38° 

37» 

36° 

' 

COSECANTS. 

802 


SECANTS. 

* 

54° 

55° 

56° 

57° 

58° 

59° 

t 

0 

1-7013016 

1-7134468 

1-7882916 

1-8360785 

1-8870799 

1-9116040 

60 

1 

1-7019831 

1-7441715 

1-7890633 

1-8369013 

1-8879589 

1-9425445 

59 

2 

1-7026653 

1-7448969 

1-7898357 

l-8377-'5l 

1-8888388 

1-9134861 

58 

3 

1-7033482 

1-7456230 

1-790G090 

1-838J498 

1  -8897197 

1-9114288 

57 

4 

1-7010318 

1-7463499 

1-7913831 

1-8393753 

1-8906016 

1-9453725 

56 

5 

1-70471(50 

1-7470776 

1-7921580 

1-8402018 

1-8914845 

1-9463173 

55 

6 

1-7054010 

1-7478060 

1-7929337 

1-8410292 

1-8923684 

1-9472632 

54 

7 

1-7060867 

1-7485352 

1-7937102 

1-8418574 

1-8932532 

1-9482102 

53 

8 

1-7067730 

1-7492651 

1-7944876 

1-8426866 

1-8941391 

1  9491583 

52 

9 

1-7074601 

1-7499958 

1-7952658 

1-8435166 

1-8950259 

1-9501075 

51 

10 

1-7081478 

1-7507273 

1-7960449 

1-8143476 

1-8059138 

1-9510577 

50 

11 

1-7088362 

1-7514595 

1-7968247 

1-8451795 

1-8968026 

1-9520091 

49 

12 

1-7095254 

1-7521924 

1-7976054 

1-8460123 

1-8976924 

1-95-29615 

48 

13 

1-7102152 

1-7529262 

1-7983S69 

1-8168460 

1-8985832 

1-9539150 

47 

14 

1-7109058 

1-7536607 

1-7991693 

1-8476806 

1-8994750 

1-9548697  . 

16 

15 

1-7115970 

1-7513959 

1-7999524 

1-8485161 

1-9003678 

1-9558254 

45 

16 

1-7122890 

1-7551320 

1-8007365 

1-8493525 

1-9012616 

1-9567822 

44 

17 

1-7129817 

l-75,ri8687 

1-8015213 

1-8501898 

1-9021564 

1-9577102 

13 

13 

1-7136750 

1-7566063 

1-8023070 

1-8510281 

1-9030522 

1-9586992 

4-2 

19 

J  -71  43691 

1-7573146 

1-8030935 

1-8518672 

1-9030491 

1-9596593 

41 

20 

1-7150639 

1-7580837 

1-8038809 

1  -8527073 

1-9018469 

1-9606206 

40 

21 

1-7157594 

1-7588236 

1-804G691 

1-8535483 

1-9057457 

1-9615329 

39 

22 

1-7164556 

1-7595642 

1-8051582 

1-8543903 

1-9066456 

1-9625464 

38 

23 

1-7171525 

1-7603057 

1-8062481 

1-8552331 

-1-9075464 

1-9635110 

37 

21 

1-7178501 

1-7610478 

1-8070388 

1-8560769 

1-9084483 

1-9644767 

36 

25 

1-7185484 

1-7617903 

1-8078304 

1-85C9216 

1-9093512 

1-9654435 

35 

26 

1-7192475 

1-7625345 

1  -8086  223 

1-8577672 

1-9102551 

1-9664114 

34 

27 

1-7199472 

1-7632791 

1-8094161 

1-8586138 

1-9111600 

1-9673805 

33 

28 

1-7206477 

1-7610244 

1-8102102 

1-8594612 

1-9120659 

1-9683507 

32 

29 

1-7213489 

1-7647704 

1-8110052 

1-8603097 

1-9129729 

1-9693220 

31 

30 

1-7220508 

1-7655173 

1-8118010 

1-8611590 

1-9138809 

1-9702944 

30 

31 

1-7227534 

1-7662619 

1-8125977 

1-8620093 

1-9147899 

1-9712680 

29 

32 

1-7231568 

1-7670133 

1-8133953 

1-86-28605 

1-9156999 

1-9722427 

28 

33 

1-7241609 

1-7677625 

1-8141937 

1-8637126 

1-9106110 

1-973-2185 

27 

31 

1-7248657 

1-7685125 

1-8149929 

1-8645657 

1-9175230 

1-9741954 

26 

35 

1-7255712 

1-7692633 

1-8157930 

1-8054197 

1-9184362 

1-9751735 

25 

36 

1-7262774 

1-7700149 

1-8165940 

1-8662747 

1-9193503 

1-9761527 

24 

37 

1-7269844 

1-7707072 

1-8173958 

1-8671306 

1-9202655 

1-9771331 

23 

38 

1-7276921 

1-7715204 

1-8181985 

1-8679875 

1-9211817 

1-9731146 

22 

39 

1-7284005 

1-7722743 

1-8190021 

1-8688453 

1-9220990 

1-9790972 

21 

10 

1-7-291096 

1-7730230 

1-3198065 

1-8699040 

1-9230173 

1-9800810 

20 

41 

1-7298195 

1-77378  15 

1-8206118 

1-8705637 

1-9239366 

1-9810659 

19 

12 

1-7305301 

1-7745409 

1-8214179 

1-8714244 

1-9248570 

1-98205-20 

18 

13 

1-7312414 

1-7752980 

1-8-22-J-249 

1-87-22859 

1-9257784 

1-9830393 

17 

11 

1-7319535 

1-7760559 

1-8-230328 

1-8731185 

1-9267009 

1-9840276 

16 

15 

1  '7526683 

1-7768146 

1-8238416 

1-8740120 

1-9276244 

1-9850172 

15 

16 

1-7333798 

1-7775741 

1-8246512 

1-8748764 

1-92854CO 

1-9860080 

14 

17 

1-7310941 

1-7783344 

1-8254017 

1-8757419 

1-9294746 

1-9869997 

13 

18 

1-7348091 

1-7790955 

1-8^-2731 

1-8766082 

1-9304013 

1-9879927 

12 

19 

1-7355248 

1-7798574 

1-8270854 

1-8774755 

1-9313290 

1-9889869 

11 

50 

1-7362413 

1-7806201 

1-8278985 

1-8783438 

'1-9322578 

1-9899822 

10 

51 

1-7369585 

1-7813836 

1-8287125 

1-8792131 

1-9331876 

1-9909787 

9 

52 

1-7376764 

1-7821479 

1-8295274 

1-8800833 

1-9341185 

1-9919764 

8 

53 

1-7383951 

1-7829131 

1-8303432 

1-8809545 

1-9350505 

1-9929752 

7 

51 

1-7391145 

1-7836790 

1-8311599 

1-8818266 

1-9359835 

1-9939753 

6 

55 

1-7398347 

1-7844457 

1-8319774: 

1-8826998 

1-9369176 

•1-9949764 

5 

56 

1-7405556 

1-7852133 

1-83-27959 

1-8835738 

1-9378527 

1-9959783 

4 

57 

1-7412773 

1-7859817 

1-8336152 

1-8844489 

1-9387889 

1-C969823 

3 

58 

1-7119997 

1-7867508 

1-8344354 

1-8858-249 

1-9397-262 

1-9979870 

2 

59 

1-7427229 

1-7875208 

1-8352565 

1-8862019 

1-9406646 

1-9989929 

1 

€0 

1-7434468 

1-7882916 

1-8360785 

1-8870799 

1-9416040 

2-0000000 

0 

/ 

35° 

34° 

>  33° 

32° 

31° 

30° 

t 

COSECANTS. 

303 


SECANTS. 

, 

60° 

61° 

62' 

63° 

64° 

65° 

' 

0 

"2-0000000 

2-0626G33 

2-1300545 

2-2026893 

2-2811720 

2-3662016 

60 

2-0010083 

2-0637484 

2-1312205 

2-2039476 

2-2825335 

2-3676787 

19 

2 

2-0020177 

2-06483-28 

2-1323830 

2-2052075 

2-2838967 

2-3691578 

>8 

3 

2-0030283 

2-0659186 

2-1335570 

2-2064691 

2-2852618 

2-3706390 

i7 

2-0040402 

2-0670056 

2-1347274 

2-2077323 

2-2866286 

2-3721222 

>6 

5 

2-0050532 

2-0680940 

2-1358993 

2-2089972 

2-2879974 

2-3736075 

55 

2-0000674 

2-0691836 

2-1370726 

2-2102637 

2-2893679 

2-3750949 

54 

7 

2-0070828 

2-0702746 

2-1382475 

2-2115318 

2-2907403 

2-3765843 

>3 

g 

2-0080994 

2-0713670 

2-1394238 

2-2128016 

2-2921145 

2-3780758 

- 

2-0091172 

2-0724606 

2-1406015 

2-2140730 

2-2934906 

2-3795694 

H 

10 

2-0101362 

2-0735556 

2-1417808 

2-2153460 

2-2948685 

2-3810650 

50 

2-0111564 

2-0746519 

2-1429615 

2-2166208 

2-2962483 

2-3S25627 

49 

12 

2-0121779 

2-0757496 

2-1441438 

2-2178971 

2-2976299 

2-3S40625 

48 

A.1 

13 

2-0132005 

2*0768486 

2-1453275 

2-2191752 

2-2990134 

2-3855645 

4* 

Ad 

14 

2-0142243 

2-0779489 

2-1465127 

2-2204548 

2-3003988 

2-3870685 

40 

15 

2-0152494 

2-0790506 

2-1476993 

2-2217362 

2-3017860 

2-3885746 

45 

16 

2-0162756 

2-0801536 

2-1488875 

2-2230192 

2-3031751 

2-3900828 

44 

17 

2-0173031 

2-0812580 

£-1500772 

2-2243039 

2-3045660 

2-3915931 

43 

18 

2-0183318 

2-0823637 

2-1512684 

2-2255903 

2-3059588 

2-3931055 

42 

19 

2-0193618 

2-0834708 

2-1524611 

2-2268783 

2-3073536 

2-3946-201 

41 

20 

2-0203929 

2-0845792 

2-1536553 

2-2281681 

2-3087501 

2-3961367 

40 

21 

2-0214253 

2-0856890 

2-1548510 

2-2294.595 

2-3101486 

2-3976555 

39 

22 

2-0224589 

2-0868002 

2-1560482 

2-2307526 

2-3115490 

2-3991764 

38 
07 

23 

2-0234937 

2-0879127 

2-1572469 

2-2320474 

2-3129513 

2-4006995 

Of 

24 

2-0245297 

2-0890265 

2-1584471 

2-2333438 

2-3143554 

2-4022247 

36 

25 

2-0255670 

2-0901418 

2-1596489 

2-2346420 

2-3157615 

2-4037520 

35 

26 

2-0266056 

2-0912584 

2-1608522 

2-2359419 

2-3171695 

2-4052815 

34 

00 

27 

2-0276453 

2-0923764 

2-1620570 

2-2372435 

2-3185794 

2-4068132 

66 

28 

2-0286863 

2-0934957 

2-1632633 

2-2385168 

2-3199912 

2-4083469 

33 

29 

2-0297286 

2-0946164 

2-1644712 

2-2398517 

2-3214049 

2-4098829 

31 

30 

2-0307720 

2-0957385 

2-1656806 

2-2411585 

2-3228205 

2-4114210 

30' 

31 

2-0318168 

2-0968620 

2-1668915 

2-2424669 

2-3242381 

2-4129613 

29 

32 

2-0328628 

2-0979869 

2-1681040 

2-2437770 

2-3256575 

2-4145038 

•7 

M 

2-0339100 

2-0991131 

2-1693180 

2-2450889 

2-3270790 

2-4160484 

** 

H 

34 

2-0349585 

2-1002408 

2-1705335 

2-2464025 

2-3285023 

2-4175952 

26 
25 

35 

2-0360082 

2-1013698 

2-1717506 

2-2477178 

2-3299276 

2-4191442 

36 

2-0370592 

2-1025002 

2-1729693 

2-2490348 

2-3313548 

2-4206954 

24 

00 

37 

2-0381114 

2-1036320 

2-1741895 

2-2503536 

2-3327840 

2-4222488 

26 

38 

2-0391649 

2-1047652 

2-1754113 

2-2516741 

2-3342152 

2-4238044 

9  22 

39 

2-0402197 

2-1058998 

2-1766346 

2-2529964 

2-3356482 

2-4253622 

21 

40 

2-0412757 

2-1070359 

2-1778595 

2-2543204 

2-3370833 

2-4269222 

20 

2-0423330 

2-1081733 

2-1790859 

2-2556461 

2-3385203 

2-4284844 

19 

42 

2-0433916 

2-1093121 

2-1803139 

2-2569736 

2-3399593 

2-4300489 

17 

43 

2-0444515 

2-1104523 

2-1815435 

2-2583029 

2-3414002 

2-4316155 

ifi 

44 

2-0455126 

2-1115940 

2-1827746 

2-2596339 

2-3428432 

2-4331844 

10 

1  r 

45 

2-0465750 

2-1127371 

2-1840074 

2-2609667 

2-3442881 

2-4347555 

Id 

46 

2-0476386 

2-1138815 

2-1852417 

2-2623012 

2-3457349 

2-4363289 

14 

1  o 

47 

2-0487036 

2-1150274 

2-1864775 

2-2636376 

2-3471838 

2-4379045 

lo 

48 

2-0497698 

2-1161748 

2-1877150 

2-2649756 

2-3486347 

2-  J  394823 

12 
II 

49 

2-0508373 

2-1173235 

2-1889541 

2-2663155 

2-3500875 

2-4410624 

in 

50 

2-0519061 

2-1184737 

2-1901947 

2-2676571 

2-3515424 

2-4426448 

IV 

51 

2-0529762 

2-1196253 

2-1914370 

2-2690005 

2-3529992 

2-4442294 

9 

<; 

52 

2-0540476 

2-1207783 

2-1926808 

2'270.-i457 

2-35445S1 

£-4458163 

53 

2-0551203 

2-1219328 

2-1939262 

2-2716927 

2-3559189 

2-4474054 

£ 

KJ 

2-0561942 

2-1230887 

2-1951733 

2-2730415 

2-3573818 

2-4189968 

04 

55 

2-0572695 

2-1242460 

2-1964219 

22743921 

2-3588467 

2-4505905 

56 

2-0583460 

2-1254048 

2-1976721 

2-2757445 

2-3603136 

2-4521865 

j 

57 

2-0594239 

2-1265651 

2-1989240 

2-2770987 

2-3617826 

2-4537848 

* 

58 

2-0605031 

2-1277267 

2-2001775 

2-2784546 

2-3632535 

2-4553853 

J 

59 

2-0615836 

2-1288899 

2-2014326 

2-2798124 

2-3647i65 

2-4569882 

j 

.  60 

2-0626653 

2-1300545 

2-2026893 

2-2311720 

2Q662016 

2-4585933 

/ 

29° 

28° 

27° 

26° 

25° 

24° 

COSECANTS. 

304 


TABLE  JLL— SECANTS  AND  COSECANTS. 


SECANTS. 


' 

66° 

67° 

68° 

69° 

70° 

71° 

2-4585933 

2-5593047 

2-6694672 

2-7904281 

2-923S044 

S-0715335 

c 

2-4602008 

2-5610599 

26713906 

2-7925444 

2-3201431 

3-0741507 

5 

24618106 

2'56i8176 

2-6733171 

2-7946641 

2-923-1858 

3-07670ij 

5 

2  -4  S  34  227 

2-5645781 

2-6752465 

2-7967873 

2-9308326 

3-0793590 

5 

2-4650371 

2-5663412 

2-6771790 

2-7989140 

2-9331833 

3-081  U702 

0 

2-4000538 

2-5681069 

2-6791145 

2-8010441 

2-9305380 

3-Obl5b60 

5 

6 

2-4682729 

2-5698752 

2-6810530 

2-8031777 

2-9378968 

3-0872066 

5 

7 

2-4693943 

2-5716162 

2-6829945 

2-8053148 

2-9402597 

3-0898319 

5 

8 

2-4715181 

2-5734199 

2-6849391 

2-8074554- 

2'9l2026j 

3-0924020 

0 

9 

2-4731442 

2-5751963 

2-6868867 

2-8095995 

2-9449975 

3-09JJP67 

5 

10 

2-1747726 

2-5769753 

2-6888374 

2-8117471 

2-9473725 

3-0977363 

C 

11 

2-4764034 

2-5787570 

2-6907912 

2-8138982 

2-9497516 

3-1003805 

4 

12 

2-4780366 

2-5805114 

2-6927480 

2-8160029 

2-9521348 

3-1030298 

4 

13 

2-4796721 

2-5823284 

2-6947079 

2-8182111 

2-9545221 

3-1050835 

4 

14 

2-4813100 

2-5841182 

2-6966709 

2-8203729 

2-9509135 

3'10d3i22 

4 

15 

2-4829503 

2-5859107 

2-6986370 

2-8225382 

2  9593090 

3-1110057 

4 

16 

2  1845929 

2-5877058 

2-7006061 

2-8247071 

2-9617087 

3-1136740 

4 

17 

2  4862380 

2-5895037 

2-7025784 

2-8268796 

2-9611125 

3-1163472 

4 

18 

2-4878854 

2-5913043 

2-7045038 

2-8290556 

2-9065205 

3-1190252 

4 

19 

2-4S95352 

2-5931077 

2-7065323 

2-8312353 

2-9089327 

3-1217081 

4 

20 

2-4911874 

2-5949137 

2-7085139 

2-8334185 

2-9713490 

3-1243959 

1 

21 

2-4528421 

2-5967225 

2-7104987 

2-8356054 

2-9737695 

3-1270S86 

3 

22 

2-4944991 

2-5985341 

2-7124866 

2-8377958 

2-9761942 

3-1297862 

3 

23 

2-4961586 

2-6003484 

2-7144777 

2-8399899 

2-9786231 

3-1321887 

3 

24 

2-4978204 

2-C021654 

2-7164719 

2-8421877 

2-9810563 

3-1351962 

3 

25 

2-4994848 

2-6039852 

2-7184693 

2-8443891 

2-983493S 

3-1379086 

3 

26 

2-5011515 

2-C05S078 

2-7204698 

2-8465941 

2-9859352 

3-1406259 

3 

27 

2-5028207 

2-6076332 

2-7224735 

2-8488028 

2-9883811 

3-1433483 

3 

28 

2-5044923 

2-6094613 

2-7244804 

2-8510152 

2-9908312 

3-1460706 

3 

29 

2-5061663 

2-6112922 

2-7261905 

2-8532312 

2-9932856 

3'148b079 

3 

30 

2-0078428 

2-6131259 

2-7285038 

2-8554510 

2-9957443 

3-1515103  . 

31 

31 

2-5095218 

2-6149624 

2-7305203 

2-8576744 

2-9982073 

3-1542877 

2C 

32 

2-5112032 

2-6168018 

2-7325400 

2-8599015 

3-0006746 

3-1570351 

2* 

33 

2-5128871 

2-6186439 

2-7345630 

2-8621324 

3-0031462 

3-1597876 

2" 

34 

2-5145735 

2-6204888 

2-7363892 

2-8643670 

-  3-0056221 

3-1625452 

it 

35 

2-5162624 

2-6223366 

2-7386186 

2-8666053 

3-0081024 

3-1653078 

23 

36 

2-5179537 

2-6241872 

2-7406512 

2-8688474 

3-0105870 

3-1680756 

24 

37 

2-5196475 

2-6260406 

2-7420871 

2-8710932 

3-0130760 

3-1708484 

23 

38 

2-5213438 

2-62789G9 

2-7447263 

2-8733428 

3-0155694 

3-1736264 

22 

39 

Atl 

2-5230426 

2-6297560 

2-7467687 

2-8755961 

3-0180672 

3-17C4095 

21 

ftv 

2-5247440 

2-5316180 

2-7488144 

2-8778J32 

8-0205693 

3-1791978 

20 

41 

2-5264478 

2-6334828 

2-7508634 

2-8801142 

3-0230759 

3-1819913 

19 

42 
43 

2-5281541 

2-6353506 

2-7529157 

2-8S23789 

3-0255868 

3-1847899 

18 

44 

2-5298630 

2-6372211 

2-7549712 

2-8846474 

3-C281023 

3-1875937 

17 

JK 

2-5315744 

2-6390946 

2-7570301 

2-8869198 

3-0306i21 

3-1904028 

16 

40 

2-5332883 

2-6409710 

2-7590923 

2-8891960 

3-0331464 

3-1932170 

15 

46 

2-5350048 

2-6428502 

2-7611578 

2-8914760 

3-0356752 

3-1960365 

14 

47 

48 

2-5367238 

2-6447323 

2-7632267 

2-8937598 

3-0382084 

3-ly8b613 

13 

49 

2-5384453 

2-6466174 

2-7652988 

2-8960475 

3-0407462 

3-2016913 

12 

r,r\ 

2-5401694 

2-6485054 

2-7673744 

2-8983391  , 

3-0432S84 

3-2045266 

U 

ou 

2-5418961 

2-6503962 

2-7694532 

2-9006346 

3-0458352 

3-2073673 

10 

51 

2-5436253 

2-6522901 

2-7715355 

2-9029339 

3-0483864 

3-2102132 

9 

£2 

53 

2-5453571 

2-6541868 

2-7736211 

2-9052372 

3-0509423 

3-2130644 

8 

2-5470915 

2-6560865 

2-7757100 

2-9075443 

3-0535026 

3-2159210 

7 

KK 

2-5488284 

2-6579891 

2-7778024 

2-9098553 

3-0560075 

3-2187830 

6 

00 

2-5505680 

2-6598947 

2-7798982 

2-9121703 

3-0086370 

3-2216503 

5 

C6 
57 

2-5523101 

2-6618033 

2-7819973 

2-9144892 

3-0012111 

3-2245230 

4 

68 

2-5540518 

2-6637148 

2-7840P99 

2-9168121 

3-01.37898 

3-  2274011 

3 

59 

2-5558022 

2-6656292 

2-7862059 

2-9191389 

3-0603731 

3-2302346 

2 

60 

2-5575521 

2-6675167 

2-7883153 

2-9214697 

3  Oi>b9610 

3  2331736 

1 

2-5593047 

2-6694672 

2-7901281 

2-9238044 

3  0715535 

3-2360680 

0 

23° 

22° 

21° 

20° 

19° 

18' 

/ 

COSEC 

ANTS. 

305 


TABLE  XI.-SECANTS  AND  COSECANTS. 


SECANTS. 

r 

72° 

73° 

74- 

75° 

76° 

77° 

0 

3-2360680 

3-4203036 

3-6279553 

3-8637033 

4-1335655 

4-4454115 

60 

1 

3-2389I.78 

3-4235611 

3-6316395 

3-8679025 

4-13*3939 

4-4510198 

59 

2 

3-2418732 

3-4263-251 

3-63533J6 

3-8721112 

4-143-2339 

4-4/J664-28 

58 

3 

3-2447840 

3-430095(5 

3-6390315 

3-8763293 

4-1180856 

4-4622803 

57 

4 

3-2477003 

3-4333727 

3-6427392 

3-8805570 

4-1529191- 

4-4679324 

56 

5 

3-2506222 

3-4366563 

3-6164548 

3-8847913 

4-1578243 

4-4735993 

55 

6 

3-2535496 

3-4399465 

3-6501783 

3-8890411 

4-16-27114 

4-4792S10 

04 

7 

3-2564825 

3-4432433 

3-6539097 

3-8932976 

4-1676102 

4-1819775 

53 

8 

3-2594211 

3-4465167 

3-6576491 

3-8975637 

4-17-20210 

4-4906889 

52 

9 

3-2623652 

3-4498368 

3-6613964 

3-9018395 

4-1774138 

4-4964152 

51 

10 

3-2653149 

3-4531735 

3-6651513 

3-9061250 

4-1823785 

4-5021065 

59 

11 

3-2682702 

3-4564969 

3-6689151 

3-9104203 

4-1873252 

4-5079129 

49 

12 

3-2712311 

3-459S269 

3-6726865 

3-9147-254 

4-1922840 

4-5136814 

48 

13 

3-2741977 

3-4631637 

3-6764660 

3-9190403 

4-1972519 

4-5194711 

47 

14 

3-2771700 

3-4665073 

3-680-2036 

3-9233651 

4-2022380 

4-5252730 

46 

15 

3-2801479 

3-4698576 

3:6840493 

3-9276997 

4-2072333 

4-5310903 

45 

16 

3-2831316 

3-4732146 

3-6878532 

3-9320443 

4-2122408 

4-5369229 

44 

17 

3-2S61209 

3-4765785 

3-6916652 

3-93U3988 

4-2172606 

4-54-27709 

43 

18 

3-2891160 

3-4799492 

3-6951854 

3-9107633 

4-2222928 

4-5186314 

42 

19 

3-2921168 

3-4833267 

3-  6993139 

3-9151379 

4-2273373 

4-5510134' 

41 

20 

3-2951234 

3-4867110 

3-7031506 

3-9495224 

4-2323943 

4-0601030 

40 

21 

3-2981357 

3-4901023 

3-7069956 

3-9539171 

4-2374637 

4-5663183 

39 

22 

3-3011539 

3-4935004 

3-7108489 

3-9583219 

4-2425457 

4-5722444 

33 

23 

3-3041778 

3-4969055 

3-7147105 

3-9627369 

4-2476402 

4-5781862 

37 

21 

3-3072076 

3-5003175 

3-7185805 

3-9671621 

4-2527474 

4-5811139 

36 

25 

3-3102432 

3-5037365 

3-7224589 

3-9715975 

4-2573671 

4-5901174 

35 

26 

3-3132347 

3-5071625 

3-7263157 

3-9760431 

4-262f>996 

4-5961070 

31 

27 

3-3163320 

3-5105954 

3-7302409 

3-9804991 

4-2681  419 

4-6021126 

33 

28 

3-3193853 

3-5140354 

3-7341446 

3-9849654 

4-2733029 

4-6081343 

32 

29 

3-3224444 

3-5174824 

3-73S0563 

3-9894,21 

4-2784733 

4-6141722 

31 

30 

3-3255095 

3-5209365 

3-7419775 

3-9939292 

4-2836576 

4-6-202263 

30 

31 

3-3285805 

3-5243977 

3-7459063 

3-9984267 

4-2888513 

4-6262967 

29 

32 

3-3316575 

3-5278660 

3-7498447 

4-0029347 

4-2940640 

4-6323835 

28 

33 

3-3347105 

3-5313414 

3-7537911 

4-0074532 

4-299!.'867 

4-6381867 

27 

31 

3-3378-294 

3-5348210 

3-7577162 

4-0119823 

4-30452-25 

4-6146061 

20 

35 

3-3409244 

3-5383138 

3-7617100 

4-0165219 

4-3097715 

4-6507427 

25 

36 

3-3440254 

3-5418107 

3-7636824 

4-0210722 

4-3150336 

4-6568956 

24 

37 

3-3471324 

3-5453149 

3-7696636 

4-0256332 

4-3203090 

4-6630652 

23 

.J8 

3-3502455 

3-5488263 

37736535 

4-0302048 

4-3255977 

4-6692516 

22 

39 

3-3533647 

3-5523450 

3-7776522 

4-0347872 

4-3308996 

4-6754518 

21 

40 

3-3564900 

3-5558710 

3-7816596 

4-0393804 

4-3362150 

4-6816718 

20 

41 

3-3596214 

3-5591042 

3-7856760 

4-0439844 

4-3415438 

4-6879119 

19 

42 

3-3627589 

3-5629448 

3-7897011 

4-0485992 

4-3468861 

4  -69  4  1600 

18 

43 

3-3659026 

3-5661928 

3-7937352 

4-0032219 

4-3522419 

4-7004372 

17 

44 

3-3690524 

3-5700481 

3-7977782 

4-0578615 

4-3576113 

4-7067256 

16 

45 

3-3722084 

3-5736108 

3-8018301 

4-0625091 

4-3629943 

4-7130313 

15 

46 

3-3753707 

3-5771810 

3-8058911 

4-0671677 

4-3683910 

4-7193542 

14 

47 

3-3785391 

3-5807586 

3-8099610 

4-0718374 

4-3738015 

4-7250945 

13 

48 

3-3817138 

3-5843137 

3-8140399 

4-0765181 

4-3792257 

4-7320521 

12 

49 

3-3848948 

3'58793(J2 

3-8181280 

4-0812100 

4-3816638 

4-7384277 

11 

50 

3-38S0820 

S'5915363 

3-8222251 

4-08591)60 

4-3901153 

4-7448206 

10 

51 

3-3912755 

3-5951439 

3-8263313 

4-090C272 

4-3955817 

4-7512312 

9 

52 

3-3944754 

3-5987590 

3-8304167 

4-0953526 

4-4010616 

4-7576596 

8 

53 

3-3976816 

3-6023818 

3-8345713 

4-1000893 

4-4065556 

4-7641028 

7 

54 

3-4008941 

3-6060121 

3-8387052 

4-1048374 

4-4120637 

4-7705699 

55 

3-4041130 

3-6096501 

3-8428482 

4-1095967 

4-4175859 

4-7770519 

56 

3-4073382 

3-6132957 

3-8470006 

4-1143675 

4-4231224 

4-7835520 

57 

3-41056'J9 

3-6169490 

3-8511622 

4-1191498 

4-4286731 

4-7900702 

58 

3-4138080 

3-6206101 

38563332 

4-1-239435 

4-4312382 

4-7966066 

59 

3-4170526 

3-624-27s8 

3-8595135 

4-1287487 

4-4398176 

4-8031613 

60 

3-4203036 

3-6279553 

3-8637033 

4-1335655 

4-4454115 

4-fcU!>?3i3 

* 

17° 

16° 

15° 

14° 

13° 

12° 

' 

COSECANTS. 

TABLE  XL— SECANTS  AND  COSECANTS. 


SECANTS. 


78° 

79° 

80° 

81° 

82° 

83° 

4-8097343 

6-2408431 

8-7587705 

6-3924532 

7-1852965 

8-2055090 

4-8163258 

5-2486979 

5-7682867 

6-4042154 

7-2001996 

8-2249952 

4-8229357 

5-2065768 

5-7778350 

6-4160216 

7-2151653 

8-2445748 

4-8295643 

5-2644798 

5-7874153 

«-4278719 

7-2301940 

8-2642485 

4-8362114 

5-2724070 

5-7970280 

6-4397666 

7-2452859 

8-2840171 

4-8428774 

5-2803587 

5-8066732 

6-4517059 

7-2604417 

8-3038812 

4-8495621 

5-2883347 

6-8163510 

6-4636901 

7-2756616 

8-3238415 

4-8562657 

5-2963354 

5-8260617 

6-4757195 

7-2909460 

8-34389S6 

4-8629883 

5-3043608 

5-8358053 

6-4877944 

7-3062954 

8-3640534 

4-8697299 

8-3124109 

5-8455820 

6-4999148 

7-3217102 

8-38430(i5 

4'8764907 

5-3204860 

5-8553921 

6-5120812 

7-3371909 

8-4046586 

4-8832707 

5-3285861 

5-8652356 

6-5242938 

7-3527377 

8-4251105 

4-8900700 

5-3367114 

5-8751123 

6-5365528 

7-3683512 

8-4456629 

4-8968886 

5-3448620 

5-8850238 

6-5488586 

7-3840318 

8-4663165 

4-9037267 

5-3530379 

5-8949688 

6-5612113 

7-3997798 

8-4S70721 

4-9105844 

5-3612393 

5-9049479 

6-5736112 

7-4155959 

8-5079304 

4-9174616 

5-3694664 

5-9149614 

6-5860587 

7-4314S03 

8-5288923 

4-9243586 

6-3777192 

6-9250095 

6-5985540 

7-4474335 

8-549D584 

4-9312754 

5-3859979 

5-9350922 

6-6110973 

7-4(534560 

8-5711295 

4-9382120 

5-3943026 

5-9452098 

6-6236S90 

7-4795482 

8-5921065 

4-9451687 

5-4026333 

5-9553625 

6-6363293 

7-4957106 

8-6137901 

4-9521453 

5-4109903 

5-9655504 

6-6490184 

7-5119437 

8-6352812 

4-9591421 

5-4193737 

5-9757737 

6-6617568 

7-5282478 

8-6568805 

4-9661591 

5-4277835 

5-9860326 

6-6740446 

7-5446236 

8-6785889 

4-9731964 

5-4362199 

5-9963274 

6-6873822 

7-5610713 

8-7004071 

4-9S02541 

5-444C831 

6-OOC6581 

6-7002699 

7-5775916 

8-7223361 

4-9873323 

5-4531731 

6-0170250 

6-7132079 

7-5941849 

8-7443766 

4-99*4311 

5-4616901 

6-0274282 

6-7261965 

7-6108516 

8-7665295 

6-0015505 

5-4702342 

6-0378680 

6-7392360 

7-6275923 

8-7887957 

5-0086907 

5-4788056 

6-0483445 

6.-7S23268 

7-6444075 

8-8111761 

6-0158517 

5-4874043 

6-0588580 

6-7654691 

7-6612976 

8-8336715 

6-0230337 

5-4960305 

6-0694085 

6-778C632 

7-6782631 

8-8562828 

6-0302367 

5-5046843 

6-0799964 

6-7919095 

7-G953047 

8-8790109 

5-0374607 

5-5133659 

6-0906219 

6-8052082 

•  7-7124227 

8-6018567 

5-0447060 

5-5220754 

6-1012850 

6-8185597 

7-7296176 

8-9248211 

5-0519726 

5-5308129 

6-1119861 

6-8319642 

7-7468901 

8-9479051 

5-0592606 

5-5395786 

6-1227253 

6-8454222 

7-7642406 

8-9711095 

5-OG65701 

5-5483726 

6-1335028 

6-8589338 

7-7816697 

8-9944354 

5-0739012 

5-5571951 

6-1443189 

6-8724995 

7-7991778 

9-0178837 

5-0812539 

•5-5660460 

6-1551736 

6-8861195 

7-816765S 

9-0414553 

5-0886284 

5-5749258 

6-1600674 

6-8997942 

-  7-8344335 

9-0651512 

6-0960248 

5-5838343 

6-1770003 

6-9135239 

7-8521821 

9-0889725 

5-1034431 

5-5927719 

6-1879725 

6-9273C89 

7-8700120 

9-1129200 

5-1108835 

5  -CO  17386 

6-1989843 

6-9111496 

7-8879238 

9-1369949 

8-1183461 

5-6107345 

6-2100359 

6-9550464 

7-9059179 

9-1611980 

5-1258309 

5-6197599 

6-2211275 

6-9689994 

7-9239950 

9-1855305 

5-1333381 

5-6288148 

6-2322.194 

6-9830092 

7-9421556 

9-2699934 

5-1408677 

5-6378995 

6-2431316 

6-9970760 

7-9604003 

9-2345877 

5-1484199 

5-6470140 

6-25JG446 

7-0112001 

7-9787298 

9-2593145 

6-1559948 

5-6561584 

6-2658984 

7-0253820 

7-9971445 

9-2841749 

5-1636924 

5-6653331 

6-2771933 

7-0396220 

8-0156150 

9-3091699 

8-1712123 

6-6745380 

6-2885295 

7-0539205 

8-0342321 

9-3343006 

5-1788563 

5-6837734 

6-29y9073 

7-0682777 

8-0529062 

9-3595B82 

5-1865228 

5-6930393 

6-5113269 

7-0826941 

8-0716681 

9-3849738 

5-19i2i25 

5-7023360 

6-3227884 

7-0971700 

8-0905182 

9-4105184 

6-2019254 

6-7116636 

6-3342923 

7-1117059 

8-1094573 

9-4362033 

5-2096618 

5-7210223 

8-3458386 

7-1263019 

8-1284860 

9-4620296 

5-2174216 

5-7304121 

6-3574276 

7-1409587 

8-1476048 

9-4879984 

5-2252050 

5-7398333 

6-3690595 

7-1556764 

8-1668145 

9-5141110 

6-2330121 

6-7492861 

6-3807347 

7-1704556 

8-1861157 

9-5403686 

5  -2408431 

5-7087705 

6-3924532 

7-1852965 

8-20SOOyQ 

9-5667722 

11° 

10° 

9° 

8° 

7° 

6° 

COSECANTS. 


TABLE  XI.-SECANTS  AND   COSECANTS. 


SECJ 

LNTS. 

' 

84° 

85° 

86° 

r     87° 

88° 

89° 

f 

9-5667722 

11-473713 

14-335587 

19-107323 

28-653708 

67-2986S8 

60 

9-5933233 

11-511990 

14-395471 

19-213970 

28-894398 

58-263755 

69 

9-6200229 

11-550523 

14-455859 

19-321816 

29-139169 

59-274308 

9-6468724 

11-589316 

14-516757 

19-430882 

29-388121 

60-314110 

67 

9-6738730 

11-628372 

14-578172 

19-541187 

29-641373 

61-391050 

5ff 

8-7010260 

11-667693 

11-640109 

19-652751 

29-899026 

62-507153 

65 

9-7283327 

11-707282 

11-702576 

19-765604 

30-161201 

63-664595 

54 

9-7557944 

11-747141 

14-765580 

19-879758 

30-128017 

64-865716 

53 

9-7834124 

11-787274 

11-829128 

19-995241 

30-699598 

66-113036 

52 

9-8111880 

11-827683 

11-893226 

20-112075 

30-976071 

67-409272 

51 

10 

9-8391227 

11-868370 

11-957882 

20-230284 

31-257577 

68-757360 

60 

11 

9-8672176 

11-909340 

15-023103 

20-349893 

31-544246 

70-160474 

49 

12 

9-8954744 

11-950595 

15-088896 

20-470926 

31-8362-25 

71-622052 

48 

13 

9-9238943 

11-992137 

15-155270 

20-593109 

32-133663 

73-145827 

47 

14 

9-9524787* 

12-033970 

15-222231 

20-717368 

.     32-436713 

74-735856 

48 

15 

9-9812291 

12-076098 

15-289788 

20-842830 

32-745537 

76-39U551 

45 

16 

10-010117 

12-118522 

15-357949 

20-969824 

33-060300 

78-132742 

44 

17 

10-039234 

12-161246 

15-426721 

21-098376 

33-381176 

79-949684 

43 

18 

10-068491 

12-204274 

15-496114 

21-228515 

33-708345 

81-853150 

42 

19 

10-097920 

12-247608 

15-566135 

21-360272 

34-041U94 

&3-849470 

41 

20 

10-127522 

12-291252 

15-636793 

21-493670 

34-382316 

85-945609 

40 

21 

10-157300 

12-335210 

15-708096 

21-628759 

34-729515 

88-149244 

39 

22 

10-187254 

12-37^484 

15-780051 

21-765553 

35-083800 

90-408863 

38 

23 

10-217386 

12-424078 

15-852676 

21-904090 

35-445391 

92-913869 

37 

21 

10-247697 

12-468995 

15-925971 

22-044403 

35-814517 

95-494711 

33 

25 

10-278190 

12-514240 

15-999918 

22-1865-J8 

36-191414 

98-223033 

35 

26 

10-308866 

12-559815 

16-074017 

22-330499 

36-576332 

101-11185 

34 

27 

10-339726 

12-605724: 

16-149987 

22-476353 

36-969528 

104-17574 

33 

28 

10-370772 

12-651971 

16-226069 

22-624126 

37-371273 

107-43114 

32 

£9 

10-402007 

12-698560 

16-302873 

22-773857 

37-781849 

110-89656 

31 

30 

10-433431 

12-745495 

16-380408 

22-9255S6 

38-201550 

114-59301 

30 

81 

10-465046 

12-792779 

16-458686 

23-079351 

38-630683 

118-54449 

29 

32 

10-496854 

12-810416 

16-537717 

23-235196 

39-069571 

122-77803 

28 

83 

10-528857 

12-888410 

16-617512 

23-393161 

39-518549 

127-32526 

27 

34 

10-561057 

12-936765 

16-098082 

23-553291 

39-977969 

132-22229 

26 

83 

10-593455 

12-985486 

16-779439 

23-715630 

40-448201 

137-51108 

25 

36 

10-626054 

13-034576 

16-861591 

23-880224 

40-929630 

143-24061 

24 

27 

10-658851 

13-084040 

16-944559 

24-047121 

41-422660 

149-46837 

23 

88 

10-691859 

13-133882 

17-028346 

24-216370 

41-927717 

150-  202  i8 

22 

89 

10-725070 

13-184106 

17-112966 

24-388020 

42  445245 

163-70325 

21 

40 

10-758488 

13-234717 

17-198431 

24-562123 

42-975713 

171-88831 

20 

41 

10-792117 

13-285719 

17-284761 

24-738731 

43-519612 

18033496 

19 

43 

10-825957 

13-337116 

17-371960 

24-917900 

44-077458 

190-98680 

18 

43 

10-860011 

13-388911 

17-460046 

25-099685 

44-649795 

202-22122 

17 

44 

10-894281 

13-141118 

17-549030 

25-234144 

45-237195 

214-85995 

16 

45 

10-928768 

H-493731 

17-638928 

25-471337 

45-840:160 

229-18385 

15 

46 

10-963476 

13-540758 

17-729753 

25-661324 

46-459625 

245-55402 

14 

47 

10-998406 

13-G00205 

17-821520 

25-854169 

47-095961 

264-44269 

13 

48 

11-033560 

13-654077 

17-914243 

26-049937 

47-749974 

2S6-47948 

12 

49 

11-068940 

13-708379 

18-007937 

26-248094 

48-422411 

312-52297 

11 

50 

11-104519 

13-763115 

18-102619 

26-450510 

49-114002 

343-77516 

10 

61 

11-140389 

13-818291 

18-198303 

26-655455 

49-825762 

381-97230 

9 

62 

11-176462 

13-873913 

18-295005 

26-863603 

50-558396 

429-71873 

8 

63 

11-212770 

13-929985 

18-392742 

27-075030 

61-312902 

491  10702 

7 

51 

11-249316 

13-986511 

18-491530 

27-289814 

62-090272 

672-95809 

6 

55 

11-286101 

11-013501 

18-591387 

27-508035 

52-891564 

687-54963 

5 

66 

11-323129 

11-100963 

18-692330 

27-729777 

63-717895 

859-43689 

4 

67 

11-360102 

11-158894 

18-794377 

27-955125 

54-570464 

1145-9157 

3 

68 

11-397922 

14-217304 

18-897545 

28-184168 

55-450531 

1718-8735 

2 

69 

11-435692 

14-276200 

19-001854 

28-416997 

56-359462 

3437-7468 

1 

60 

11-473713 

14-335587 

19-107323 

28-653708 

67-298688 

Infinite. 

0 

' 

6" 

4° 

3° 

2° 

1° 

0° 

7 

COSE 

CANTS. 

TABLE  XII.— TANGENTS  AND  COTANGENTS. 


0° 

1°            li            2°                         3° 

S 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang      Tang     Cotang 

0 

.00000 

Infinite. 

.01746 

57.2900 

.03492 

28.6363 

.05241 

19.0811 

60 

1 

.00029 

3437.75 

.01775 

56.3506 

.03521 

28.3994 

.05270 

18.9755 

59 

2 

.00058 

1718.87 

.01804 

55.4415 

.03550 

28.1664 

.05299 

18.8711  |58 

3 

.00087 

1145.92 

.01833 

54.5613 

.03579 

27.9372 

.05328 

18.7678    57 

4 

.00116 

859.436 

.01862 

53.7086 

.03609 

27.7117 

.05357 

18.6656    56 

5 

.00145 

687.549 

.01891 

52.8821 

.03638 

27.4899 

.05387 

18.5645    55 

0 

.00175 

572.957 

.01920 

52.0807 

.03667 

27.2715 

.05416 

18.4645 

54 

7 

.00204 

491.106 

.01949 

51.3032 

.03696 

27.0566 

.05445 

18.3655 

53 

S 

.00233 

429.718 

.01978 

50.5485 

.03725 

26.8450 

.05474 

18.2677 

52 

g 

.00262 

381.971 

.02007 

49.8157 

.03754 

26'.  6367 

.05503 

18.1708 

51 

10 

.00201 

343.774 

.02036 

49.1039 

.03783 

26.4316 

.05533 

18.0750 

50 

11 

.00320 

312.521 

.02066 

48.4121 

.03812 

26.2296 

.05562. 

17.9802 

49 

12 

.00349 

286.478 

.02095 

47.7395 

.03842 

26.0307 

.05591 

17.8863 

48 

13 

.00378 

264.441 

.02124 

47.0853 

.03871 

25.8348 

.05620 

17.7934 

47 

14 

.00407 

245.552 

.02153 

46.4489 

.03900 

25.6418 

.05649 

17.7015 

46 

15 

.00436 

229.182 

.02182 

45.8294 

.03929 

25.4517 

.05678 

17.6106 

45 

16     .00465 

214.858 

.02211 

45.2261 

.03958 

25.2644 

.05708 

17.5205 

44 

17     .00495 

202.219 

.02240 

44.6386 

.03987 

25.0798 

.05737 

17.4314 

43 

18  |   .00534 

190.984 

.02269 

44.0661 

.04016 

24.8978 

.05766 

17.3432 

42 

1!) 

.00553 

180.932 

.02298 

43.5081 

.04046 

24.7185 

.05795 

17.2558 

41 

SO 

.00582 

171.885 

.02328 

42.9641 

.04075 

24.5418 

.05824 

17.1693 

40 

21 

.00611 

163.700 

.02357 

42.4335 

.04104 

24.3675 

.05854 

17.0837 

39 

22 

.00640 

156.259 

.02386 

41.9158 

.04133 

24.1957 

.05883 

16.9990 

38 

2:5 

.00609 

149.465 

.02415 

41.4106 

.04162 

24.0263 

.05912 

16.9150 

37 

2! 

.00698 

143.237 

.02444 

40.9174 

.04191 

23.8593 

.05941 

16.8319 

36 

25 

.00727 

137.507 

.02473 

40.4358 

.04220 

23.6945 

.05970 

16.7496 

35 

96 

.00756 

132.219 

,02502 

39.9655 

.04250 

23.5321 

.05999 

16.6681 

34 

27 

.00785 

127.321 

.02531 

39.5059 

.04279 

23.3718 

.06029 

16.5874 

33 

28 

.00815 

122.774 

.02560 

39.0568 

.04308 

23.2137 

.06058 

16.5075 

32 

;».,' 

.00844 

118.540 

.02589 

38.6177 

.04337 

23.0577 

.06087 

16.4283 

31 

yu 

.00873 

114.589 

.02619 

38.1885 

.04366 

22.9038 

.06116 

16.3499 

30 

31 

.00902 

110.892 

.02648 

37.7680 

.04395 

22.7519 

.06145 

16.2722 

29 

82 

.00931 

107.426 

.02677 

37.3579 

.04424 

22.6020 

.06175 

16.1952 

28 

83 

.00960 

104.171 

.02706 

36.9560 

.04454 

22.4541 

.06204 

16.1190 

27 

34 

.00989 

101.107 

.02735 

36.5627 

.04483 

22.3081 

.06233 

16.0435 

26 

88 

.01018 

98.2179 

.02764 

38.1776 

.04512 

22.1640 

.06262 

15.9687 

25 

86 

.01047 

95.4895 

.02793 

35.8006 

.04541 

22.0217 

.06291 

15.8945 

24 

a; 

.01076 

92.9085 

.02822 

35.4313 

.04570 

21.8813 

.06321 

15.8211 

23 

as 

.01105 

90.4633 

.02851 

35.0695 

.04599 

21.7426 

.06350 

15.7483 

22 

89 

.01135 

88.1436 

.02881 

34.7151 

.04628 

21.6056 

.06379 

15.6762 

21 

40 

.01164 

85.931)8 

.02910 

34.3678 

.04658 

21.4704 

.06408 

15.6048 

20 

41 

.01193 

83.8435 

.02939 

34.0273 

.04687 

21.3369 

.06437 

15.5340 

19 

•12 

.01222 

81.8470 

.02968 

33.6935 

.04716 

21.2049 

.06467 

15.4638 

18 

13 

.01251 

79.9434 

.02997 

33.3662 

.04745 

21.0747 

.06496 

15.3943 

17 

41 

.01280 

78.1263 

.03026 

33.0452 

.04774 

20.9460 

.06525 

15.3254 

10 

45 

.01309 

76.3900 

.03055 

32.7303 

.04803 

20.8188 

.06554 

15.2571 

15 

10 

.€1338 

74.7292 

.03084 

32.4213 

.04833 

20.6932 

.06584 

15.1893 

14 

47 

.01367 

73.1390 

.03114 

32.1181 

.04862 

20.5691 

.06613 

15.1222 

13 

48 

.01396 

71.6151 

.03143 

31.8205 

.04891 

20.4465 

.06642 

15.0557 

12 

40 

.01425 

70.1533 

.03172 

31.5284 

.04920 

20.3253 

.06671 

14.9898 

11 

50 

.01455 

68.7501 

.03201 

31.2416 

.04949 

20.2056 

.06700 

14.9244 

10 

51 

.01484 

67.4019 

.03230 

30.9590 

.04978 

20.0872 

.06730 

14.8596 

9 

52 

.01513 

66.1055 

.03259 

30.6833 

.05007 

19.9702 

.06759 

14.7954 

8 

68 

.01542 

64.8580 

.03288 

30.4116 

.05037 

19.8546 

.06788 

14.7317 

T 

51 

.01571 

63.6567 

.03317 

30.1446  1 

.05086 

19.7403 

.06817 

14.6685 

6 

55 

.01600 

62.4992 

.03346 

29.8823 

.05095 

19.6273 

.06847 

14.6059 

5 

66 

.01629 

61.3829 

.0.3376 

29.6245 

.05124 

19.5156 

.06876 

14.5438 

4 

57 

.01658 

60.3058 

.03405 

29.3711 

.05153 

19.4051 

.06905 

14.4823 

3 

58 

.01687 

59.2659 

.03434 

29.1220 

.05182 

19.2959 

.06934 

14.4212 

2 

60 

.01716 

58.2612 

.03463 

28.8771 

.05212 

19.1879 

.06963 

14.3607 

1 

60 

.01746 

57.2900 

.03492 

28.6363 

.05241 

19.0811 

.06993 

14.3007 

0 

f 

Cotangj    Tang 

Cotang 

Tang      Cotang 

Tang 

Cotang  j    Tang 

f 

89° 

88°           I            87° 

86° 

309 


TABLE  XII.— TANGENTS  AND  COTANGENTS. 


40                  50 

6°                          7° 

Tang     Cotang 

Tang     Cotang 

Tang     Cotang      Tang   ;  Cotang 

' 

0 

.06993 

14.3007 

.08749 

11.4301 

.10510     9.51436       .12278  i  8.14485 

00 

1     .07022 

14.2411 

.08778 

11.3919 

.10540     9.48781  I 

.12308 

8.12481 

59 

2     .07051 

14.1821  i 

.08807 

11.3540 

.10569 

9.46141  1 

.12338 

8.10536 

58 

3     .07U80 

14.1235  j 

.08837 

11.3163 

.10599 

9.43515 

.12367 

8.08600 

57 

4     .07110 

14.0655  ! 

.08866 

11.2789 

.10628 

9.40904 

.12397 

8.06674 

5(> 

5     .07139 

14.0079  i 

.08895 

11.2417 

.10657 

9.38307  , 

.12426 

8.04756 

55 

6    .07168 

13.9507  1 

.08925 

11.2048 

.10687 

9.35724 

.12456 

8.02848 

54 

r* 

.07197 

13.8940 

.08954 

11.1681 

.10716 

9.33155 

.12485 

8.00948 

53 

8 

.07227 

13.8378  ! 

.08983 

11.1316 

.10746 

9.30599 

>J2515 

7.99058 

52 

9 

.07256 

13.7821 

.09013 

11.0954 

.10775 

9.28058 

.12544 

7.97176 

51 

10 

.07285 

13.7267 

.09042 

11.0594 

.10805 

9.25530 

.12574 

7.95302 

50 

11 

.07314 

13.6719 

.09071 

11.0237 

.10834 

9.23016 

.12608 

7.93438 

49 

12 

.07344 

13.6174 

.09101  , 

10.9882 

.10863 

9.20516 

.12633 

7.91582 

-IS 

13 

.07373 

13.5634 

.09130 

10.9529 

.10893 

9.18028 

.12662 

7.89734 

47 

14     .07402 

13.5098 

.09159 

10.9178 

.10922 

9.15554 

.12692 

7.87895 

46 

15     .07431 

13.4566 

.09189 

10.8829 

.10952 

9.13093 

.12722 

7.86064 

45 

16     .07461 

13.4039 

.09218 

10.8483 

.10981 

9.10646 

.12751 

7.84242 

44 

17 

.07490 

13.3515 

.09247 

10.8139 

.11011 

9.08211 

.12781 

7.82428 

48 

18 

.07519 

13.2996 

.09277 

10.7797 

.11040 

9.05789 

.12810 

7.80622 

42 

19 

.07548 

13.2480 

.09306 

10.7457 

.11070 

9.03379 

.12840 

7.78825 

41 

20 

.07578 

13.1969 

.09335 

10.7119 

.11099 

9.00983 

.12869 

7.77035 

40 

21 

.07607 

13.1461 

.09365 

10.6783 

.11128 

8.98598 

.12899 

7.75254 

39 

22 

.07636 

13.0958 

.09394 

10.6450 

.11158 

8.96227 

.12929 

7.73480 

m 

23 

.07665 

13.0458 

.09423 

10.6118 

.11187 

8.93867 

.12958 

7.71715 

37 

24 

.07695 

12.9962 

.09453 

10.5789 

.11217 

8.91520 

.12988 

7.69957 

86 

25 

.07724 

12.9469 

.09482 

10.5462 

.11246 

8.89185 

.13017 

7.68208 

36 

2(i 

.07753 

12.8981 

.09511 

10.5136 

.11276 

8.86862 

.13047 

7.66466 

34 

27 

.07782 

12.8496 

.09541 

10.4813 

.11305 

8.84551 

.13076 

7.64732 

m 

28 

.07812 

12.8014 

.09570 

10.4491 

.11335 

8.82252 

.13106 

7.63005 

32 

29 

.07841 

12.7536 

.09600 

10.4172 

.11364 

8.79964 

.13136 

7.61287 

81 

30 

.07870 

12.7062 

.09629 

10.3854 

.11394 

8.77689 

.13165 

7.59575 

30 

31 

.07899 

12.6591 

.09658 

10.3538 

.11423 

8.75425 

.13195 

7.57872 

29 

32 

.07929 

12.6124 

.09688 

10.3224    i    .11452 

8.73172 

.13224 

7.56176 

28 

33    .07958 

12.5660 

.09717 

10.2913    i    .11482 

8.70931 

.13254 

7.54487 

27 

34 

.07987 

12.5199 

.09746 

10.2602  1     .11511 

8.68701 

.13284 

7.52806 

20 

35 

.08017 

12.4742 

.09776 

10.2294  i     .11541 

8.66482 

.13313 

7.51132 

•.T, 

36 

.08046 

12.4288 

.09805 

10.1988 

.11570 

8.64275 

.13343 

7.49465 

24 

37 

.08075 

12.3838 

.09834 

10.1683 

.11600 

8.62078 

.13372 

7.47806 

23 

88 

.08104 

12.3390 

.09864 

10.1381 

.11629 

8.59893 

.13402 

7.46154 

•>•> 

39 

.08134 

12.2946 

.09893 

10.1080 

!    .IK    i 

8.57718 

.13432 

7.44509 

•21 

40 

.08163 

12.2505 

.09923 

10.0780 

.11688 

8.55555 

.13461 

7.42871 

20 

41 

.08192 

12.2067 

.09952 

10.0483 

!   .11718 

P.  53402 

.13491 

7.41240 

19 

42 

.08221 

12.1632 

.09981 

10.0187 

!    .11747 

8.51259 

.13521 

7.39616 

18 

43 

.08251 

12.1201 

.10011 

9.98931 

8.49128 

.13550 

7.37999 

17 

44 

.08280 

12.0772 

.10040 

9.96007 

.'11806 

8.47007 

.13580 

7.36389 

16 

45 

.08309 

12.0346 

.10069 

9.93101 

.11836 

8.44896 

.13609 

7.34786 

15 

46 

.08339 

11.9923 

.10099 

9.90211 

.11865 

8.42795 

.13639 

7.33190 

14 

47 

.08368 

11.9504 

.10128 

9.87338 

.11895 

8.40705 

.13669 

7.31600 

18 

48 

.08397 

11.9087 

.10158 

9.84482 

.11924 

8.38625 

.13698 

7.30018 

12 

49 

.08427 

11.8673 

.10187 

9.81641 

.11954 

8.36555 

.13728 

7.28442 

11 

50 

.08456 

11.8262 

.10216 

9.78817 

.11983 

8.34496 

.13758 

7.26873 

10 

51 

.08485 

11.7853 

.10246 

8.76009 

.12013 

8.32446 

.13787 

7.25310 

9 

52 

.08514 

11.7448 

.10275 

9.73217 

.12042 

8.30406 

.13817 

7.23754 

8 

53 

.08544 

11.7045 

.10305 

9.70441 

.12072 

8.28376 

.13846 

7.22204 

7 

54 

.08573 

11.6645 

.10334 

9.67680 

.12101 

8.26355 

.13876 

7.20661 

6 

55 

.08602 

11.6248 

.10363 

9.64935 

.12131 

8.24345 

.13906 

7.19125 

5 

56 

.08632 

11.5853 

.10393 

9.62205 

.12160 

8.22344 

.13935 

7.17594 

4 

57 

.08661 

11.5461 

.10422 

9.59490 

.12190 

8.20352 

.13965 

7.16071 

8 

58 

.08690 

11  5072 

.10452 

9.56791 

.12219 

8.18370 

.13995 

7.14553 

2 

59 

.08720 

11.4685 

.10481 

9.54106       .12249 

8.16398 

.14024 

7.13042 

1 

CO 

.08749 

11.4301 

.10510 

9.51-436       .12278     8.14435 

.14054 

7.11537 

0 

/ 

Cotang 

Tang 

Cotang 

Tang      Cotang     Tang 

Cotang 

Tang 

/ 

85° 

84°           il            83° 

82° 

310 


TABLE  XII.-TANGENTS  AND  COTANGENTS. 


8°      -      i             9° 

10°           li           11° 

/ 

Tang  |  Cotang  |  Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

o 

.14054 

7.11537 

.15838 

6.31375 

.17633 

5.67128 

.19438 

5.14455 

GO 

1 

.140,84 

7.10038 

.15868 

6.30189 

.17663 

5.66165 

.19468 

5.13658 

59 

2 

.14113 

7.08546 

.15898 

6.29007 

.17693 

5.65205 

.19498 

5.12862 

58 

3 

.14143 

7.07059 

.15928 

6.27829 

.17723 

5.64248 

.19529 

5.12069 

57 

4 

.14173 

7.05579 

.15958 

6.20655 

.17753 

5.63295 

.19559 

5.11279 

56 

5 

.14202 

7.04105 

.15988 

6.25486 

.17783 

5.62344 

.19589 

5.10490 

55 

6 

.14232 

7.02637 

.16017 

6.24321 

.17813 

5.61397 

.19619 

5.09704 

54 

7 

.14262 

6.91174 

.16047 

6.23160 

.17843 

5.60452 

.19649 

5.08921 

58 

8 

.14291 

6.99718 

.16077 

6.22003 

.17873 

5.59511 

.19680 

5.08139 

68 

9 

.14321     6.98268 

.16107 

6.20851 

.17903 

5.58573 

.19710 

5.07360 

51 

10 

.14351 

6.96823 

.16137 

6.19703 

.17933 

5.57638 

.19740 

5.06584 

50 

11 

.14381 

6.95385 

.16167 

6.18559 

.17963 

5.56706 

.19770 

5.05809 

49 

12 

.14410 

6.93952 

.16196  !  6.17419 

.17993 

5.55777 

.19801 

5.05037 

48 

13 

.14440 

6.92525 

.16226 

6.16283 

.18023 

5.54851 

.19831 

5.042G7 

47 

14 

.14470 

6.91104 

i   .16256 

ft  15151 

.18053 

5.53927 

.19861 

5.03499 

46 

15 

.14499 

6.89688 

.16286 

6.14023 

.18083 

5.53007 

.19891 

5.02734 

45 

16 

.14589 

6.88278 

.16316 

6.12899 

.18113 

5.52090 

.19921 

5.01971 

44 

17 

.14559 

6.86874 

.16346 

6.11779 

.18143 

5.51176 

.19952 

5.01210 

43 

18 

.14588 

6.85475 

.16376 

6.10664 

.18173 

5.50264 

.19982 

5.00451 

42 

19 

.14618 

6.84082 

.16405 

6.09552 

.18203 

5.49356 

.20012 

4.99695 

41 

20 

.14648 

6.82694 

.16435 

6.08444 

.18233 

5.48451 

.20042 

4.98940 

40 

21 

.14678 

6.81312 

.16465 

6.07340 

.18263 

5.47548 

.26073 

4.98188 

39 

22 

.  14707 

6.79936 

.16495 

6.06240 

.18293 

5.46648 

.20103 

4.97438 

38 

23 

.14737 

6.7'8564 

.16525 

6.05143 

.18323 

5.45751 

.20133 

4.96690 

37 

24 

.14767 

6.77199 

.16555 

6.04051 

.18353 

5.44857 

.20164 

4.95945 

36 

25 

.14796 

6.75838 

.16585 

6.02962 

.18384 

5.43966 

.20194 

4.95201 

35 

26 

.14826 

6.74483 

.16615 

6.01878 

.18414 

5.43077 

.20224 

4.94460 

34 

27 

.14856 

6.73133 

.16645 

6.00797 

.18444 

5.42192 

.20254 

4.93721 

33 

28 

.14886 

6.71789 

.16674 

5.99720 

.18474 

5.41309 

.20285 

4.92984 

32 

29 

.14915 

6.70450 

.16704 

5.98646 

.18504 

5.40429 

.20315 

4.92249 

31 

30 

.14945 

6.69116 

.16734 

5.97576 

.18534 

5.39552 

.20345 

4.91516 

30 

81 

.14975 

6.67787 

.16764 

5.96510 

.18564 

5.38677 

.20376 

4.90785 

29 

32 

.15005 

6.66463 

.16794 

5.95448 

.18594 

5.37805 

.20406 

4.90056 

28 

33 

.15034 

6.65144 

.16824 

5.94390 

.18624 

5.36936 

.20436 

4.89330 

27 

34 

.15064 

6.63831 

.16854 

5.93335 

.18654 

5,36070 

.20466 

4.88605 

26 

35 

.15094 

6.62523 

.16884 

5.92283 

.18684 

5.35206 

.20497 

4.87882 

25 

36 

.15124 

6.61219 

.16914 

5.91236 

.18714 

5.34345 

.20527 

4.87162 

24 

37 

.15153 

6.59921 

.16944 

5.90191 

.18745 

5.33487 

.20557 

4.86444 

23 

38 

.15183 

6.58627 

.16974 

5.89151 

.18775 

5.32631 

.20588 

4.85727 

22 

39 

.15213 

6.57339 

.17004 

5.88114 

.18805 

5.31778 

.20618 

4.85013 

21 

40 

.15243 

6.56055 

.17033 

5.87080 

.18835 

5.30928 

.20648 

4.84300 

20 

41 

.15272 

6.54777 

.17063 

5.86051 

.18865 

5.30080 

.20679 

4.83590 

19 

42 

.15302 

6.53503 

.17093 

5.85024 

.18895 

5.29235 

.20709 

4.82882 

18 

43 

.15332 

6.52234 

.17123 

5.84001 

.18925 

5.28393 

.207'39 

4.82175 

17 

44 

.15362 

6.50970 

.17153 

5.82982 

.18955 

5.27553 

.20770 

4.81471 

16 

45 

.15391 

6.49710 

.17183 

5.81966 

.18986 

5.26715 

.20800 

4.80769 

15 

46 

.15421 

6.48456 

.17213 

5.80953 

.19016 

5.25880 

.20830 

4.80068 

14 

47 

.15451 

6.47206 

.17243 

5.79944 

.19046 

5.25048 

.20861 

4.79370 

13 

48 

.15481 

6.45961 

.17273 

5.78938 

.19076 

5.24218 

.20891 

4.78673 

12 

49 

.15511 

6.44720 

.17303 

5.77936 

.19106 

5.23391 

.20921 

4.77978 

11 

50 

.15640 

6.43484 

.17333 

5.76937 

.19136 

.5.22566 

.20952 

4.77286 

10 

51 

.15570 

6.42253 

.17363 

5.75941 

.19166 

5.21744 

.20982 

4.76595 

9 

52 

.15600 

6.41026 

.17393 

5.74949 

.19197 

5.20925 

.21013 

4.75906 

8 

53 

.15630 

6.39804 

.17423 

5.73960 

.19227 

5.20107 

.21043 

4.75219 

7 

54 

.15660 

6.38587 

.17453 

5.72974 

.19257 

5.19293 

.21073 

4.74534 

6 

55 

.15689 

6.37374 

.17483 

5.71992 

.19287 

5.18480 

.21104 

4.73851 

5 

56 

.15719 

6.36165 

.17513 

5.71013 

.19317 

5.17671 

.21134 

4.73170 

4 

57 

.15749 

6.34961 

.17543 

5.70037 

.19347 

5.16863 

.21164 

4.72490 

8 

58 

.15779 

6.33761 

.17573 

5.69064 

.19378 

5.16058 

.21195 

4.71813 

2 

59 

.15809 

6.32566 

.17603 

5.68094  [I    .19408 

5.15256 

.21225 

4.71137 

1 

60 

.isass 

6.31375 

.176:33 

5.67128 

.19438 

5.14455 

.21256 

4.70463 

0 

f 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

/ 

81°                       80° 

79° 

78° 

311 


TABLE  XII.— TANGENTS  AND  COTANGENTS. 


12° 

13° 

14° 

15° 

Tang 

Cotang 

Tang 

Cotang 

Tang  i  Cotang 

Tang 

Cotang 

i 

0 

.21256 

4.70463 

.23087     4.33148 

.24933  j  4.01078 

.26795  i  3.73205 

60 

1 

.21286 

4.69791 

.23117  |  4.32573 

.24964  ;  4.00582 

.26826  !  3.72771 

59 

a 

.21316 

4.69121 

.23148 

4.32001 

.24995  i  4.00086 

.26857     3.72338 

58 

3 

.21347 

4.68452 

.23179 

4.31430 

.25026     3.99592 

.26888  ;  3.71907 

57 

4 

.21377 

4.67786 

.23209 

4.30860 

.25056     3.99099 

.26920  i  3.71476 

56 

5 

.21408 

4.67121 

.23240 

4.30291 

.25087     3.98607 

.26951 

3.71046 

55 

G 

.21438 

4.66458 

.23271 

4.29724 

.25118     3.98117 

.26982 

3.7'0616 

54 

7 

.21469 

4.65797 

.23301 

4.29159 

.25149 

3.97627 

.27013 

3.70188 

53 

8 

.21499 

4.65138 

.23332 

4.28595 

.25180 

3.97139 

.27044 

3.69761 

52 

9 

.215£9 

4.64480 

.23363 

4.28032 

.25211 

3.96651 

.27076 

3.69a35 

51 

10 

.21560 

4.63825 

.23393 

4.27471 

.25242 

3.96165 

.27107 

3.68909 

50 

11 

.21590 

4.63171 

.23424 

4.26911 

.25273 

3.95680 

.27138 

3.68485 

49 

12 

.21621 

4.62518 

.23455 

4.26352 

.25304     3.95196 

.27169 

3.68061 

48 

13 

.21651 

4.61868 

.23485 

4.25795 

.25335      3.94713 

.27201 

3.67638 

47 

14 

.21682 

4.61219 

.23516 

4.25239 

.25366     3.94232 

.27232 

3.67'217 

46 

15 

.21712 

4.60572 

.23547 

4.24685 

.25397      3.937ol 

.27263 

3.66796 

45 

16 

.21743 

4.59927 

.23578 

4.24132 

.25428  ;  3.93271 

.27294 

3.66376 

44 

17 

.21773 

4.59283 

.23608 

4.23580 

.25459     3.92793 

.27326 

3.65957 

43 

18 

.21804 

4.58641 

.23639 

4.23030 

.25490     3.92316 

.27357 

3.65538 

42 

19 

.21834 

4.58001 

.23670 

4.22481 

.25521 

3.91839 

.27388 

3.65121 

41 

20 

.81864 

4.57363 

.23700 

4.21933 

.25552 

3.91364 

.27419 

3.647'05 

40 

21 

.21895 

4.56726 

.23731 

4.21387 

.25583  !  3.90890 

.27451 

3.64289 

39 

22 

.21925 

4.56091 

.23762 

4.20842 

.25614  ]  3.90417 

.27482 

3.63874 

38 

23 

.21956 

4.55458 

.23793 

4.20298 

.25645     3.89945 

.27513 

3.63461 

37 

21 

.21986 

4.54826 

.23823 

4.19756 

.25676      3.89474 

.27545 

3.63048 

36 

25 

.52017 

4.54196 

.23854 

4.19215 

.25707 

3.89004 

.27576 

3.62636 

35 

26 

.22047 

4.53568 

.23885 

4.18675 

.25738 

3.88536 

.27607 

3.62224 

34 

27 

.22078 

4.52941 

.23916 

4.18137 

.25769 

3.88068 

.27638 

3.61814 

33 

28 

.22108 

4.52316 

.23946 

4.17600 

.25800  :  3.87601 

.27670 

3.61405 

32 

29 

.22139 

4.51693 

.23977 

4.17064 

.25831   I  3.87136 

.27701 

3.60996  !31 

30 

.22169 

4.51071 

.24008 

4.16530 

.25862 

3.86671 

.27732  |  3  60588    30 

31 

.22200 

4.50451 

.24039 

4.15997 

.25893 

3.86208 

.27764  1  3.60181    29 

32 

.22231 

4.49832 

.24069 

4.15465 

.25924 

3.85745 

.27795  i  3.59775 

28 

33 

.22261 

4.49215 

.24100 

4.14934 

.25955 

3.85284 

.27826 

3.59370 

27 

34 

.22292 

4.48600 

.24131 

4.14405 

.25986 

3.84824 

.27858 

3.58966 

26 

85 

.22322 

4.47986 

.24162 

4.13877 

.26017 

3.84364 

.27889 

3.58562 

25 

86 

.22353 

4.47374 

.24193 

4.13350 

.26048 

3.83906 

.27921 

3.58160 

24 

87 

.22383 

4.46764 

.24223 

4.12825 

26079  ;  3.83449 

.27952 

3.57758 

23 

38 

.22414 

4.46155 

.24254 

4.12301 

.26110  i  3.82992 

.27983 

3.57357 

22 

39 

.22444 

4.45548 

.24285 

4.11778 

.26141 

3.82537 

.28015 

3.56957 

21 

40 

.22475 

4.44942 

.24316 

4.11256 

.26172 

3.82083 

.28046 

3.56557 

20 

41 

.22505 

4.44338 

.24347 

4.10736 

.26203 

3.81630 

.28077 

3.56159 

19 

42 

.22536 

4.43735 

.24377 

4.10216 

.26235  !  3.81177 

.28109 

3.55761 

18 

43 

.22567 

4.43134 

.24408 

4.09699 

.26266  !  3.80726 

.28140 

3.55364 

17 

44 

.22597 

4.42534 

.24439 

4.09182 

.26297 

3.80276 

.28172 

3.r>l%8 

16 

45    .22628 

4.41936 

.24470 

4.08666 

.26328 

3.79827 

.28203 

3.54573 

15 

46  j   .22658 

4.41340 

.24501 

4.08152 

.26359 

3.79378 

.28234 

3.54179 

14 

47    .22689 

4.40745 

.24532 

4.07639 

.26390 

3.78931 

.28266 

3.53785 

13 

48  1   .22719 

4.40152 

.24562 

4.07127 

.26421 

3.78485 

.28297 

3.53393 

12 

49 

.22750 

4.39560 

.24593 

4.06C16 

.26452 

3.78040 

.28329 

,3.53001 

11 

50 

.22781 

4.38969 

.24624 

4.06107 

.20183 

3.77595 

.28360 

3.52609 

10 

51 

.22811 

4.38381 

.24655 

4.05599 

.26515 

3.77152 

.28391 

3.52219 

9 

5-3 

.22842 

4.37793 

.24686 

4.05092 

.26546     3.76709 

.28423 

3.51829 

8 

63 

.22872 

4.37207 

.24717 

4.04586 

.26577  i  3.76268 

.28454 

3.51441 

7 

r>4 

.22903 

4.36623 

.24747 

4.04081 

.26608     3.75828 

.28486 

3.51053 

6 

55 

.22934 

4.36040 

.24778 

4.03578 

.26639 

3.75388 

.28517 

3.50666 

*5 

f,<5 

.22964 

4.35459 

.24809 

4.03076 

.26670 

3.74950 

.28549 

3.50279 

4 

57 

.22995 

4.34879 

.24840 

4.02574 

.26701 

3.74512 

.28580 

3.49894 

3 

58 

.23026 

4.34300 

.24871 

4.02074 

.26733 

3.74075 

.28612 

3.49509 

2 

59 

.23056 

4.33723 

.24902 

4.01576 

.26764 

3.73640 

.28043 

3.49125 

1 

60 

.23087 

4.33148 

.24933 

4.01078 

.26796 

3.73205 

.28675 

3.48741 

0 

j 

Cotang 

Tang 

Cotang 

Tang 

Cotang     Tang 

Cotang     Tang 

7 

77° 

76° 

T5° 

74° 

313 


TABLE  xn.— TANGENTS  AND  COTANGENTS. 


16°      '                 17° 

18° 

19° 

Tang 

Cotang 

Tang   j  Cotang 

Tang 

Cotang 

Tang 

Cotang 

o 

3.48741 

.30573 

3.27085 

.32492 

3.07768 

.34433" 

2.90421    60 

1     .28706     3.48359 

.30605 

3.26745 

.32524 

3.07464 

.34465 

2.90147 

59 

2     .287:38     3.47977 

.30637 

3.26406 

.32556 

3.07160 

.34498 

2.89873 

58 

3     .28769     3.47596 

.30669 

3.26067 

.32588 

3.06857 

.34530 

2.89600 

57 

4     .28800     3.47216 

.30700 

3.25729 

.32621 

3.06554 

.34563 

2.89327 

5<> 

f> 

.28832     3.46837 

.30732 

3.25392 

.32653 

3.06252 

.34596 

2.89055 

55 

6 

.28864      3.46458 

.30764 

3.25055 

.32685 

3.05950 

.34628 

2.88783 

54 

7    .28895     3.46080 

.30796 

3.24719 

.32717 

3.05649 

.34661 

2.88511 

53 

8i   .28927 

3.45703 

.30828 

3.24383 

.32749 

3.05349 

.34693 

2.88240 

52 

9'   .28958 

3.45327 

.30860 

3.24049 

.32782 

3.05049 

.34726 

2.87970 

51 

10  |   .28990 

3.44951 

.30891 

3.23714 

.32814 

3.04749 

.34758 

2  87700 

50 

11 

.29021 

3.44576 

.30923 

3.23381 

.32846 

3.04450 

.34791 

2.87430 

49 

12 

.29053 

3.44202 

.30955 

3.23048 

.32878 

3.04152 

.34824 

2.87161 

48 

13;   .29084 

3.43829 

.30987  . 

3.22715 

.32911 

3.03854 

.34856 

2.86892 

47 

14    .29116 

3.43456 

.31019 

3.22384 

.32943 

3.03556 

.34889 

2.86624 

46 

15!   .29147 

3.43084 

.31051 

3.22053 

.32975 

3.03260 

.34922 

2.86356 

45 

16    .29179 

3.42713 

.31083 

3.21722 

.33007 

3.02963 

.34954 

2.86089 

44 

17 

.29210 

3.42343 

.31115 

3.21392 

.33040 

3.02667 

.34987 

2.85822 

43 

18 

.29242 

3.41973 

.31147 

3.21063 

.33072 

3.02372 

.35020 

2.85555 

42 

19 

.29274 

3.41604 

.31178 

3.20734 

.33104 

3.02077 

.35052 

2.85289 

41 

20 

.29305 

3.41236 

.31210 

3.20406 

.33136 

3.01783 

.35085 

2.85023 

40 

Oj 

.29337 

3.40869 

.31242 

3.20079 

.33169 

3.01489 

.8S1HB 

2.84758 

39 

28 

.29368 

3.40502 

.31274 

3.19752 

.33201 

3.01196 

.35150 

2.84494 

38 

23 

.29400 

3.40136 

.31306 

3.19426 

.33233 

3.00903 

.35183 

2.84229 

87 

24 

.29432 

3.39771 

.31338 

3.19100 

.33266 

3.00611 

.35216 

2.&S96:. 

86 

25 

.29463 

3.39406 

.31370 

3.18775 

.33298 

3.00319 

.35248 

2.83702 

35 

28 

.29495 

3.39042 

.31402 

3.18451 

.33330 

3.00028 

.35281 

2.83439 

34 

27 

.29526 

3.38679 

.31434 

3.18127 

.33363 

2.99738 

.35314 

2.83176 

88 

28     .21)558 

3.38317 

.31466 

3.17804 

.33395 

2.99447 

.35346 

2.82914 

32 

29 

.29590 

3.37955 

.31498 

3.17481 

.33427 

2.99158 

.35379 

2.82653 

31 

30 

.29621 

3.37594 

.31530 

3.17159 

.33460 

2.98868 

.35412 

2.82391 

30 

33 

.29653 

3.37234 

.31562 

3.16838 

.33492 

2.98580 

.35445 

2.82130 

29 

32 

.29685 

3.36875 

.31594 

3.10517 

.33524 

2.98292 

.35477 

2.81870 

28 

88 

.29716 

3.3G516 

.31626 

3.16197 

.a3557 

2.98004 

.35510 

2.81610 

27 

:J1 

.29748 

3.36158 

.31658 

3.15877 

.33589 

2.97717 

.35543 

2.81350 

•M 

38 

.29780 

3.35800 

.31690 

3.15558 

.33621 

2.97430 

.35576 

2.81091 

25 

36 

.29811 

3.35443 

.31722 

3.15240 

.33654 

2.97144 

.35608 

2.80833 

24 

37    .29843 

3.35087 

.31754 

3.1492.2 

.33686 

2.96858 

.35641 

2.80574 

23 

88!   .29875 

3.34732 

.31786 

3.14605 

.33718 

S.  96573 

.35674 

2.80316 

22 

89 

.29906 

3.34377 

.31818 

3.14288 

.33751 

2.96288 

.35707 

2.80059 

21 

W 

.29938 

3.34023 

.31850 

3.13972 

.33783 

2.96004 

.35740 

2.79802 

96 

41 

.29970 

3.33670 

.31882 

3.13656 

.33816 

2.95721 

.35772 

2.79545 

19 

42    .30001 

3.33317 

.31914 

3.13341 

.33848 

2.95437 

.35805 

2.79289 

18 

43 

.30033 

3.32965 

.31946 

3.13027 

.33881 

2.95155 

.35838 

2.79033 

17 

11 

.30065 

3.32614 

.31978 

3.12713 

.33913 

2.94872 

.35871 

2.78778 

1(5 

45 

.30097 

3.32264 

.32010 

3.12400 

,aS945 

2.94591 

.35904 

2.78523 

15 

46    .30128 

3.31914 

.32042 

3.12087 

.33978 

2.94309 

.35937 

2.78269 

14 

47!   .30160 

3.31565 

.32074 

3.11775 

.34010 

2.94028 

.35969 

2.78014 

13 

18     .30192 

3.31216 

.32106 

3.11464 

.34043 

2.93748 

.36002 

2.77761 

12 

•I!)     .:!0-."J4 

3.30868 

.32139 

3.11153 

.34075 

2.93468 

.36035 

2.77507 

11 

50 

.30255 

3.30521 

.32171 

3.10842 

.34108 

2.93189 

.36068 

2.77254 

10 

51 

.30287 

3.30174 

.32203 

3.10532 

.34140 

2.92910 

.36101 

2.77002 

9 

58 

.30319 

3.29829 

.32235 

3.10223 

.34173 

2.92632 

.36K34 

2.76750 

8 

:>:>, 

.30351 

3.29483 

.32267 

3.09914 

.34205 

2.92354 

.36167 

2.76498 

7 

54 

'  .30382 

3.29139 

.32299 

3.09606 

.34238 

2.92076 

.36199 

2.76247 

6 

55 

.30414 

3.28795 

.32331 

3.09298 

.34270 

2.91799 

.36232 

2.75996 

5 

58 

.30446 

3.28452 

.32363 

3.08991 

.34303 

2.91523 

.36265 

2.75746 

4 

57 

.30478 

3.28109 

.32396 

3.08685 

.343:35 

2.91246 

.36298 

2.75496 

3 

58 

.30509 

3.27767 

.32428 

3.08379 

.34368 

2.90971 

.38381 

2.75246 

2 

59 

.30541 

3.27426 

.32460 

3.08073 

.34400 

2.90696 

.36364 

2.74997 

1 

60 

.30573 

3.27085 

.32492 

3.07768 

.34433 

2.90421 

.36397 

2.74748 

0 

/ 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

/ 

73° 

72° 

71°           II           70° 

—  A 

BIB 


TABLE  XIL— TANGENTS  AND  COTANGENTS. 


20°            !            21°            ! 

22°           i            23° 

Tang   |  Cotang 

Tang 

Cotang 

Tang 

Cotang  '    Tang 

Cotang 

0    .36397  |  2.74748 

!   .38386 

2.60509 

.40403 

2.47509 

.42447  |  2.35585 

60 

1     .33430     2.74499 

.38420 

2.602&3  ! 

.40436 

2.47302 

.42482 

2.35395 

59 

2    .36463  I  2.74251 

.38453 

2.60057 

.40470 

2.47095 

.42516 

2.35205 

58 

3    .36496 

2.74004 

.38487 

2.59831 

.40504 

2.46888    !   .42551 

2.35015 

57 

4    .36529 

2.73756 

.38520 

2.59606 

.40538 

2.46682 

.42585 

2.34825 

56 

5    .36562 

2.73509 

.38553 

2.59381 

.40572 

2.46476 

.42619 

2.34636 

55 

6    .36595 

2.73263 

.38587 

2.59156 

.40606 

2.46270 

.42654 

2.34447 

54 

7    .36628 

2.73017 

.38620 

2.58932 

.40640 

2.46065 

.42688 

2.34258 

58 

8    .36661 

2.72771 

.38654 

2.58708 

.40674 

2.45860 

42722 

2.34069 

52 

9     .36694 

2.72526 

.38687 

2.58484 

.40707 

2.45655 

'.42757 

2.33881 

51 

10-1    .36727 

2.72281 

.38721 

2.58261 

.40741 

2.45451 

.42791 

2.33693 

50 

11 

.36760 

8.72036 

.38754 

2.58038 

.40775 

2.45246 

.42826 

2.33505 

49 

12 

.36793 

2.71792 

.38787 

2.57815 

.40809 

2.45043 

.42860 

2.33317 

18 

13 

.36826 

2.71548 

.38821 

2.57593 

.40843 

2.44839 

.42894 

2.33130 

47 

14 

.36859 

2.71305 

.38854 

2.57371 

.40877 

2.44636 

.42929 

2.32943 

4G 

15 

.36892 

2.71062 

.38888 

2.57150 

.40911 

2.44433 

.42963 

2.32756 

45 

16 

.36925 

2.70819 

.38921 

2.56928 

.40945 

2.44230 

.42998 

2.32570 

44 

17 

.36958 

2.70577 

.38955 

2.56707 

.40979 

2.44027 

.43032 

2.32383 

43 

18 

.36991 

2.70335 

.38988 

2.56487 

.41013 

2.43825 

.43067 

2.32197 

42 

19 

.37024 

2.70094 

.39022 

2.56266 

.41047 

2.43623 

.43101 

2.32012 

41 

20 

.37057 

2.69853 

.39055 

2.56046 

.41081 

2.43422 

.43136 

2.31826 

10 

21 

.37090 

2.69612 

.39089 

2.55827 

.41115 

2.43220 

.43170 

2.31641 

39 

22 

.37123 

2.69371 

.39122 

2.55608 

.41149 

2.43019 

.43205 

2.31456  138 

.37157 

2.69131 

.39156 

2.55389 

.41183 

2.42819 

.43239 

2.31271 

37 

24 

.37190 

2.68892 

.39190 

2.55170 

.41217 

2.42618 

.43274 

2.31086 

86 

25 

.37223 

2.68653 

.39223 

2.54952 

.41251 

2.42418 

.43308 

2.30902 

35 

26 

.37256 

2.68414 

.39257 

2.54734 

.41285 

2.42218 

.43343 

2.30718 

34 

27 

.37289 

2.68175 

.39290 

2.54516 

.41319 

2.42019 

.43378 

2.30534 

33 

28 

.37322 

2  67937 

.39324 

2.54299 

.41353 

2.41819 

.43412 

2.30351 

3-3 

2'J 

.37355 

2.67700 

.39357 

2.54082 

.41387 

2.41620 

.43447 

2.30167 

31 

30 

.37388 

2.67462 

.39391 

2.53865 

.41421 

2.41421 

.43481 

2.29984 

30 

31 

.37422 

2.67225 

.39425 

2.53648 

.41455 

2.41223 

.43516 

2.29801 

20 

32 

.37455 

2.66989 

.39458 

2.53432 

.41490 

2.41025 

.43550 

2.29619 

28 

33 

.37488 

2.66752 

.39492 

2.53217 

.41524 

2.40827 

.43585 

2.29437 

27 

34 

.37521 

2.66516 

.39526 

2.53001 

.41558 

2.40G29 

.43620 

2.29254 

26 

35 

.37554 

2.66281 

.39559 

2.52786 

.41592 

2.40432 

.43654 

2.29073 

25 

36 

.37588 

2.66046 

.39593 

2.52571 

.41626 

2.40235 

.43689 

2.28891 

21 

37 

37621 

2.65811 

.39626 

2.52357 

.41660 

2.40038 

.43724 

2.28710 

23 

38 

.37654 

2.65576 

.39660 

2.52142 

.41694 

2.39841 

.43758 

2.28528 

go 

39 

37687 

2.65342 

.39694 

2.51929 

.41728 

2.39645 

.43793 

2.28348 

21 

40 

.37720 

2.65109 

.39727 

2.51715 

.41763 

2.39449 

.43828 

2.28167 

20 

41 

.37754 

2.64875 

.39761 

2.51502 

.41797 

2.39253 

.43862 

2.27987 

19 

42 

.37787 

2  64642 

i     39795 

2.51289 

.41831 

2.39058 

.43897 

2.27806 

18 

43 

.37820 

2.64410 

i   .39829 

2.51076 

.41865 

2.38863 

.43933 

2.27626 

17 

44 

37853 

2.64177 

!   .39862 

2.50864 

41899 

2.38668 

.43966 

2.27447 

16 

45 

.37887 

2.63945 

i   .39896 

2.50652 

.41933 

2.38473 

.44001 

2.27267 

15 

46 
47 

.37920 
.37953 

2.63714 
2.63483 

.39930 
.39963 

2.50440 
2.50229 

.41968 
.42002 

2.38279 
2.38084 

.44036 
.44071 

2.27088 
2.26909 

11 
13 

48 

.37986 

2.63252 

i   .39997 

2.50018 

.42036 

2.37891 

.44105 

2.26730 

12 

49 

38020 

2.63021 

!    .40031 

2.49807 

.42070 

2.37697 

.44140 

2.26552 

11 

50 

.38053 

2.62791 

i    .40065 

2.49597 

.42105 

2.37504 

.44175 

2.26374 

10 

51 

38086 

2.62561 

.40098 

2.49386 

.42139 

2.37311 

.44210 

2.26196 

9 

52 

.38120 

2.62332 

.40132 

2.49177 

.42173 

2.37118 

.44244 

2.26018 

H 

53 

.38153 

2.62103 

.40166 

2.48967 

.42207 

2.36925 

.44279 

2.25840 

7 

54 

.38186 

2.61874 

.40200     2.48758 

.42242 

2.36733 

.44314 

2.25663 

6 

55 

38220 

2.61646 

.40234 

2.48549 

.42276 

2.36541 

.44349 

2.25486 

5 

56 
57 

.38253 
38286 

2.61418 
2.61190 

.40267 
.40301 

2  48340 

2.48132 

.42310 
.42345 

2.36349 
2.36158 

.44384 
.44418 

2.25309 
2.25132 

4 

3 

58 

.38320 

2.60963 

.40335      2.47924 

.42379 

2.35967 

.44453     2.24956 

2 

59 
90 

.38353 
.38386 

2.60736 
2.60509 

.40369     2.47716 
.40403     2.47509 

.42413 

.42447 

2.35776 
2  35585 

.44488     2  24780 
.44523     2.24604 

1 
0 

Cotang 

Tang 

Cotang      Tang     h  Cotang      Tang 

Cotang     Tang 

, 

1            69° 

68° 

67°          ii           66° 

314 


TABLE  m— TANGENTS  AND  COTANGENTS. 


24°        'I!           25°                       26°            !           27° 

Tang  j  Cotang 

Tang 

Cotang      Tang 

Cotang 

Tang 

Cotang 

0 

T44523 

2.24604 

.46631" 

2.14451   ;     .48773 

2.05030 

.50953 

1.96261    60 

1 

.44558 

2.24428 

.46666 

2.14288    1    .48809 

2.04879 

.50989 

1.96120  '59 

8 

.44593 

2.24252 

.46702 

2.14125 

i   .48845 

2.04728 

.51026 

1.95979    58 

8 

.44627 

2.24077 

.46737 

2.13963 

j   .48881 

2.04577 

.51063 

1.95838  157 

4 

.44662 

2.23902 

.46772 

2.13801 

.48917 

2.04426 

.51099 

1.95698  '56 

5 

.44697 

2.23727 

.46808 

2.13639 

!   .48953 

2.04276 

.51136 

1.95557    55 

6 

.44732 

2.23553 

.46843 

2.1:3477 

I   .48989 

2.04125 

.51173 

1.95417    54 

7 

.44767 

2.23378 

.46879 

2.13316 

.49026 

2.03975 

.51209 

1.95277    53 

8 

.44802 

2.23204 

.46914 

2.13154 

!   .49062 

2.03825 

.51246 

1.95137 

52 

9 

.44837 

2.23030 

.46950 

2.12993 

i   .49098 

2.03675 

.51283 

1.94997 

51 

10 

.44872 

2.22857 

.46985 

2.12832 

.49134 

2.03526 

.51319 

1.94858 

50 

11 

.44907 

2.22683 

'   .47021 

2.12671 

.49170 

2.03376 

.51356 

1.94718 

49 

12 

.44942 

2.22510 

.47056 

2.12511 

.49206 

2.03227 

.51393 

1.94579 

48 

13 

.44977 

2.22337 

.47092 

2.12350 

.49242 

2.03078 

.51430 

1.94440 

47 

14 

.45012 

2.22164 

.47128 

2.12190 

.49278 

2.02929 

.51467 

1.94301 

46 

15 

.45047 

2.21992 

.47163 

2.12030 

.49315 

2.02780 

.51503 

1.94162 

t5 

16 

.45082 

2.21819 

.47199 

2.11871 

.49351 

2.02631 

.51540 

1.94023    44 

17 

.45117 

2.21647 

.472:34 

2.11711 

.49387 

2.02483 

.51577 

1.93885  ;43 

18 

.45152 

2.21475 

.47-270 

2.11552 

.49423 

2.02335 

.51614 

1.93746  |42 

11) 

.45187 

2.21304 

.47305 

2.11392 

.49459 

2.02187 

.51651 

1.93608  141 

20 

.45222 

2.21132 

.47341 

2.11233 

.49495 

2.02039 

.51688 

1.93470 

40 

21 

.45257 

2.20961 

.47377 

2.11075 

.49532 

2.01891 

.51724 

1.93332 

39 

22 

.45292 

2.20790 

.€r412 

2.10916 

.49568 

2.01743 

.51761 

1.93195 

38 

23 

.45327 

2.20619 

.47448 

2.10758 

.49604 

2.01596 

.51798 

1.93057 

37' 

24 

.45362 

2.20449 

.47483 

2.10600 

!    .49640 

2.01449 

.51835 

1.92920    36 

25 

.45397 

2.20278 

.47519 

2.10442 

!    .49677 

2.01302 

.51872 

1.92782 

35 

26 

.45432 

2.20108 

.47555  •  2.10284 

i    .49713 

2.01155 

.51909 

1.92645 

34 

27 

.454(57 

2.19938 

.47590  !  2.10126 

!    .49749 

2.01008 

.51946 

1.92508 

33 

28 

.45502 

2.19769 

.47626 

2.09969 

!   .49786     2.00862 

.51983 

1.92371 

3-2 

29 

.45538 

2.19599 

.47062 

2.09811 

.49822  :  2.00715 

.52020 

1.92235 

31 

30 

.45573 

2.19430 

.47698 

2.09654 

.49858     2.00569 

.52057 

1.92098 

30 

31 

.45608 

2.19261 

.47733 

2.09498 

.49894 

2.00423 

.52094 

1.91962 

29 

3-2 

.45643 

2.19092 

.47769 

2.09341 

!   .49931 

2.00277 

.52131 

1.91826 

28 

33 

.45678 

2.18923 

.47805 

2.09184 

.49967 

2.00131 

.52168 

1.91690 

27 

34    .45713 

2.18755 

.47840 

2.09028 

.50004 

1.99986 

.52205 

1.91554 

26 

35    .45748 

2.18587 

.4787'6 

2.08872 

.50040 

1.99841 

.52242 

1.91418 

25 

36     .45784 

2.18419 

.47912 

2.08716 

.500J'6 

1.99695 

.52279 

1.91282 

•24 

37    .45819 

2.18251 

.47948      2.08560 

.50113 

1.99550 

.52316 

1.91147 

•23 

38    .45854 

2.18084 

.47984  !  2.0S405 

.50149 

1.99406 

.52353 

1.91012 

22 

39     .45889 

2.17916 

.48019      2.08250 

.50185 

1.99261 

.52390 

1.90876    21 

40 

.45924 

2.17749 

.48055 

2.08094 

.50222 

1.99116 

.52427 

1.90741 

20 

41 

.45960 

2.17582 

.48091 

2.07939 

.50258 

1.98972 

.52464 

1.90607 

19 

42 

.45995 

2.17416 

.48127 

2.07785 

.50295 

1.98828 

.52501 

1.90472 

18 

43     .46030 

2.17249 

.48163 

2.07630 

.50331      1.98684 

.52538 

1.90337 

17 

44     .46065 

2.17083 

.48198 

2.07476 

|   .50368     1.98540 

.52575 

1.90203 

1C 

45!    .46101 

2.16917 

.48234 

2.07321 

:    .50404  !  1.98396 

.52613 

1.90069 

15 

46    .46136  !  2.16751 

.48270 

2.07167 

i    .50441 

1.98253 

.52650 

1.89935  114 

47    .46171 

2.16585 

.48306 

2.07014 

i   .50477 

1.98110 

.52687 

1.8S801  Sl3 

48    .46206 

2.16420 

.48342 

2.06860 

i   .50514 

1.97'966 

.52724 

1.89667    12 

49!   .46242 

2.16255 

.48378 

2.067'06 

!    .50550 

1.97823 

.52761 

1.895:33 

11 

50     .46277 

2.16090 

.48414 

2.06553 

i   .50587 

1.97681 

.52798 

1.89400 

10 

51 

.46312 

2.15925 

.48450 

2.06400 

.50623 

1.97538 

.52836 

1.8926b 

9 

52 

.46348 

2.15760 

.48486     2.06247 

.50660 

1.97395 

.52873 

1.89133 

8 

53 

.46383 

2.15596 

.48521 

2.0G094 

.50696 

1.97253 

.52910 

1.89000 

7 

54 

.46418 

2.15432 

.48557 

2.05942 

.50733 

1.97111 

.52947 

1.88867 

6 

55     .46454 

2.15268 

.48593 

2.05790 

.50769 

1.96969 

.52985 

1.88734 

5 

56     .46489 

2.15104 

148629 

2.05637 

.50806 

1.96827 

.53022 

1.88602 

4 

57  i   .46525 

2.14940 

.48665 

2.05485 

.50843 

1.96685 

.53059 

1.88469 

3 

58!   .46560 

2.14777 

.48701 

2.05333 

.50879 

1.96544 

.53096 

1.  88337 

2 

59     .46595 

2.14614 

.48737 

2.05182 

.50916 

1.96402 

.53134 

1.88205 

1 

60 

.46631 

2.14451 

.48773  '  2.05030 

.50953 

1.96261 

.53171 

1.88073 

0 

/ 

Cotang 

Tang 

Cotang  \    Tang 

Cotang 

Tang 

Cotang 

Tang 

r 

65° 

64° 

;           63° 

62° 

315 


TABLE  XII.-TANGENTS  AND  COTANGENTS. 


2 

8° 

2 

9° 

3 

0°  

3 

1° 

/ 

Tang 

Cotang 

1   Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang- 

/ 

0 

.53171 

.88073 

.55431 

1.80405 

.57735 

1.73205 

.60086 

1.66428 

60 

1 

.53208 

.87941 

.55469 

1.80281 

.57774 

1.73089 

.60126 

1.66318 

59 

2 

.53246 

.87809 

.55507 

1.80158 

.57813 

1.72973 

.60165 

1.66209 

58 

3 

.53283 

.87677 

.55545 

1.80034 

.57851 

1.72857 

.60205 

1.66099 

57 

4 

.53320 

.87546 

.55583 

1.79911 

.57890 

1.72741 

.60245 

1.65990 

56 

5 

.53358 

.87415 

.55621 

1.79788 

.57929 

1.72625 

.602S4 

1.65881 

55 

G 

.53395 

.87283 

.55659 

1.79665 

.57968 

1.72509 

.60324 

1.65772 

54 

7 

.53432 

.87152 

.55697 

1.79542 

.58007 

1.72393 

.60364 

1.65663 

58 

8 

.53470 

.87021 

.55736 

1.79419 

.58046 

1.72278 

.60403 

1.65554 

52 

9 

.53507 

.86891 

.55774 

1.79296 

.58085 

1.72163 

.60443 

1.65445 

51 

10 

.53545 

.86760 

.55812 

1.79174 

:   .58124 

1.72047 

.60483 

1.65337 

50 

11 

.53582 

.86630 

.55850 

1.79051 

.58162 

1.71932 

.60522 

1.65228 

-10 

13 

.53620 

.86499 

.55888 

1.78929 

1   .58201 

1.71817 

.60562 

1.65120 

48 

13 

.53657 

.86369 

.55926 

1.78807 

.58240 

1.71702 

.60602 

1.65011 

47 

14 

.53694 

.86239 

.55964 

1.78685 

.58279 

1.71588 

.60642 

1.64903 

46 

15 

.53732 

.86109 

.56003 

1.78563 

.58318 

1.71473 

.60681 

1.64795 

45 

16 

.53769 

.85979 

.56041 

1.78441 

.58357 

1.71358 

.60721 

1.64687 

44 

17 

.53807 

.85850 

.56079 

1.78819 

i   .58396 

1.71244 

.60761 

1.64579 

48 

18 

.53844 

.85720 

.56117 

1.78198 

.58435 

1.71129 

.60801 

1.64471 

42 

10 

.53882 

.85591 

.56156 

1.78077 

;   .58474 

1.71015 

.60841 

1.64363 

41 

20 

.53920 

.85462 

.56194 

1.77955 

.58513 

1.70901 

.60881 

1.64256 

40 

21 

.53957 

.85333 

.56232 

1.77834 

1   .58552 

1.70787 

.60921 

1.64148 

89 

22 

.53995 

.85204 

.56270 

1.77713 

.58591 

1.70673 

.60060 

1.64041 

88 

23 

.54032 

.85075 

.56309 

1.77592 

.58631 

1.70560 

.61000 

1.  63934 

87 

24 

.54070 

.84946 

.56347 

1.77471 

.58670 

1.70446 

!   .61040 

1.63826 

36 

25 

.54107 

.84818 

.56385 

1.77351 

.58709 

1.70332 

i    .61080 

1.63719 

35 

2G 

.54145 

.84689 

.56424 

1.77230 

.58748 

1.70219 

1   .61120 

1.63612 

31 

27 

.54183 

.84561 

.56462 

1.77110 

.58787 

1.70106 

.61160 

1.63505 

33 

28 

.54220 

.84433 

.56501 

1.76990 

.58826 

1.69992 

.61200 

1.63398 

32 

2!) 

.54258 

.84305 

.5(5539 

1.76869 

.58865 

1.69879 

.61240 

1.63292 

31 

30 

.54296 

,84177 

.56577 

1.76749 

.58905 

1.69766 

.61280 

1.63185 

80 

31 

.54333 

.84049 

.56616 

1.76629 

.58944 

1.69653 

!    .61320 

1.63079 

29 

32 

.54371 

.83922 

.56654 

1.76510 

.58988 

1.61)541 

j    .61360 

1.62972 

28 

33 

.54409 

.a3794 

.56693 

1.76390 

;   .59022 

1.69428 

.61400 

1.62866 

27 

34 

.54446 

.83667 

.56731 

1.76271 

.59061 

1.69316 

.61440 

1.62760 

26 

35 

.54484 

.83540 

.56769 

1.76151 

.59101 

1.69203 

.61480 

1.62654 

25 

36 

.54522 

.83413 

.56808 

1.76032 

.59140 

1.69091 

.61520 

1.62548 

24 

37 

.5-1560 

.83286 

.56846 

1.75913 

.59179 

1.68979 

.61561 

1.62442 

23 

38 

.54597 

.83159 

.56885 

1.75794 

:   .59218 

1.68866 

.61601 

1.62336 

22 

3'J 

.54635 

.83033 

.56923 

1.75675 

.59258 

1.68754 

.61641 

1.62230 

21 

40 

.54673 

.82906 

.56962 

1.75556 

.59297 

1.68643 

.61681 

1.62125 

20 

41 

.54711 

.82780 

.57000 

1.75437 

.59386 

1.68531 

.61721 

1.62019 

1!) 

42 

.54748 

.82654 

.57039 

1.75319 

i   .59376 

1.68419 

.61761 

1.61914 

18 

43 

.54786 

.82528 

.  57078 

1.75200 

.59415 

1.68308 

.61801 

1.61808 

17 

44 

.54824 

.82402 

.57116 

1.75082 

.59454 

1.68196 

.61842 

1.61703 

16 

45 

.54862 

.82276  i 

.57155 

1.74964 

.59494 

1.68085 

.61882 

1.61598 

15 

46 

.54900 

.82150  1 

.57193 

1.74846 

!   .59533 

1.67974 

.61922 

1.61493 

14 

47 

.54938 

.82025 

.57232 

1.74728 

.59573 

1.67863 

.61962 

1.61388 

13 

48 

.54975 

.81899 

.57271 

1.74610 

.59612 

1.67752  , 

.62003 

1.61283 

12 

4!» 

.55013 

.81774 

.57309 

1.74492 

.59651 

1.67641 

.68043 

1.6117!) 

11 

no 

.55051 

.81649 

.57348 

1.74375 

.59691 

1.67530 

.62083 

1.61074 

10 

51 

,55089 

.81524 

.57386 

1.74257 

.59730 

1.67419 

.62124 

l.fiOflTO 

9 

52 

.55127 

.81399 

.57425 

1.74140 

;   .59770 

1.67309 

.62164 

1.60865 

8 

53 

.55165 

1.81274 

.57464 

1.74022 

|   .59809 

1.67198 

.62204 

1.60761 

7' 

54 

.55203 

1.81150 

.57503 

1.73905 

!   .59849 

1.67088 

.62245 

1.60657 

6 

55 

.55241 

1.81025 

.57541 

1.73788 

.59888 

1.66978 

.62285 

1.60553 

5 

56 

.55279 

1.80901  l 

.57580 

1.73671 

.59928 

1.66867 

.62325 

1.60449 

4 

57 

.55317 

1.80777 

.57619 

1.73555 

.59967 

1.66757 

.62366 

1.60345 

8 

58 

.55355 

L80658 

.57657 

1.73438 

.60007 

1.66647 

.62406 

1.60241 

2 

59 

.55393 

1.80529 

.57696 

1.78821 

.60046 

1.66538 

.62446 

1.60137 

1 

GO 

.55431 

1.80405  | 

.57735 

1.73205 

.60086 

1.66428 

.62487 

1.60033 

0 

/ 

Cotaug 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tung 

( 

6 

1°             1 

6 

0° 

5 

5° 

5 

8° 

316 


TABLE  XII.— TANGENTS  AND  COTANGENTS. 


32°      .     i            33°           j            34°                        35° 

Tang     Co  tang  '     Tang 

Cotang   | 

Tang     Cotang      Tang 

Cotang 

0     .t  1:2187      1.6003:-] 

.64941 

1.53986 

.07451 

1.48256 

.70021 

1.42815 

60 

1     .625-27      1.59930 

.64982 

1.53888 

.67493 

1.48163 

.70064 

1.42726 

59 

2    .62568 

1.59826 

.65024 

1  .53791 

.67536 

1.48070 

.70107 

1.42638 

58 

3     .62608 

1.59723 

.650(15 

1.53693 

.67578 

1.47977 

.70151 

1.42550 

57 

4     .62649 

1.59620 

.65106 

1.53595 

.67620 

1.47885 

.70194 

1.42462 

56 

5 

.62689 

1.59517 

.65148 

1.53497 

.67663 

1.47792 

.70238 

1.42374 

55 

6 

.62730 

1.59414 

.65189 

1.53400 

.67705 

1.47699 

.70281 

1.42286 

54 

y 

.62770 

1.59311 

.65231 

1.53302 

.67748 

1.47607 

.70325 

1.42198 

53 

8 

.62811 

1.59208 

.65272 

1.53205 

.67790 

1.47514 

.70368 

1.42110 

52 

9 

.62852 

1.59105 

.65314 

1.53107 

.67832 

1.47422 

.70412 

1.42022 

51 

10 

.62892 

1.59002 

.65355 

1.53010 

.67875 

1.47330 

.70455 

1.41934 

50 

11 

.62933 

1.58900 

.65397 

1.52913 

.67917 

1.47238 

.70499 

1.41847 

49 

12 

.62973 

1.58797 

.65438 

1.52816 

.67960 

1.47146 

.70542 

1.41759 

48 

13 

.63014 

1.58695 

.65480 

1.52719 

.68002 

1.47053 

.70586 

1.41672 

47 

14 

.63055 

1.58593 

.65521 

1.52622 

.68045 

1.46962 

.70629 

1.41584 

46 

15 

.63095 

1.58490 

.65563 

1.52525 

.68088 

1.46870 

.70673 

1.41497 

45 

6 

.63136 

1.58388 

.65604 

1.52429 

.68130 

1.46778 

.70717 

1.41409 

44 

.63177 

1.58286 

.65646 

1.52332 

.68173 

1.46686 

.7'0760 

1.41322 

43 

8 

.63217 

1.58184 

.65688 

1.52235 

.68215 

1.46595 

.70804 

1.41235 

42 

19 

.63258 

1  .58083 

.65729 

1.52139  : 

.68258 

1.46503 

.70848 

1.41148 

41 

20 

.63299 

1.57981 

.65771 

1.52043 

.68301 

1.46411 

.70891 

1.41061 

40 

21 

.63340 

1.57879 

.65813 

1.51946  ' 

.68343 

1.46320 

.70935 

1.40974 

39 

22 

.63380 

1.57778 

.65851 

1.51850  I 

.68386 

1.46229 

.70979 

1.40887 

38 

23 

.63421 

1.57676 

.65896 

1.51754  i 

.68429 

1.46137 

.71033 

1.40800 

37 

24     .63462 

1.57575 

.65938 

1.51658 

.68471 

1.46046 

.71066 

1.40714 

36 

25 

.63503 

1.57474 

.65980 

1.51562 

.68514 

1.45955 

.71110 

1.40627 

35 

26 

.63544 

1.57372 

.66021 

1.51466 

.68557 

1.45864 

.71154 

1.40540 

34 

27 

.63584 

1.57271 

.66063 

1.51370 

.68600 

1.45773 

.71198 

1.40454 

33 

23 

.63625 

1.57170 

.66105 

1.51275 

.68642 

1.45682 

.71242 

1.40367 

32 

2'J 

.63666 

1.57069 

.66147 

1.51179 

.68685 

1.45592 

.71285 

1.40281 

31 

30 

.63707 

1.56969 

.66189 

1.51084 

.68728 

1.45501 

.71329 

1.40195 

30 

31 

.63748 

1.56868 

.66230 

1.50988 

.68771 

1.45410 

.71373 

1.40109 

29 

32 

.63789 

1.56767 

.66272 

1.50893 

.68814 

1.45320 

.71417 

1.40022 

28 

33 

.63830 

1.56667 

.66314 

1.50797 

.68857 

1.45229 

.71461 

1.39936 

27 

34 

.63871 

1.56566 

.66356 

1.50702 

.68900 

1.45139 

.71505 

1.39850 

26 

35 

.63912 

1.56466 

.66398 

i;  50607 

.68942 

1.45049 

.71549 

1.39764 

25 

36 

.63953 

1.56366 

.66440 

1.50512 

.68985 

1.44958 

.71593 

1.39679 

24 

37 

.63994 

1.56265 

.66482 

1.50417 

.69028 

1.44868 

.71637 

1.39593 

23 

38 

.64035 

1.56165 

.66524 

1.50322 

.69071 

1.44778 

.71681 

1.39507 

22 

39 

.64076 

1.56005 

.66566 

1.50228 

.69114 

1.44688 

.71725 

1.39421 

21 

40 

.64117 

1.55966 

.66608 

1.50133 

.69157 

1.44598 

.71769 

1.39336 

20 

41 

.64158 

1.55866 

.66650 

1.50038 

.69200 

1.44508 

.71813 

1.39250 

19 

42 

.64199 

1.55766 

.66692 

1.49944 

.69243 

1.44418 

.71857 

1.39165 

18 

43 

.64240 

1.55666 

.66734 

1.49849 

.69286 

1.44329 

.71901 

1.39079 

17 

44 

.64281 

1.55567 

.66776 

1.49755 

.69329 

1.44239 

.71946 

1.38994 

16 

45 

.64322 

1.55467 

.66818 

1.49661 

.69372 

1.44149 

.71990 

1.38909 

15 

46 

.64363 

1.55368 

.66860 

1.49566 

.69416 

1.44060 

.72034 

1.38824 

14 

47 

.64404 

1.55269 

'   .66902 

1.49472 

.69459 

1.43970 

.72078 

1.38738 

13 

48 

.64446 

1.55170 

!   .66944 

1.49378 

.69502 

1.43881 

.72122 

1.38653 

12 

49 

.64487 

1.55071 

i    .66986 

1.49284 

.69545 

1.43792 

.72167 

1.38568 

11 

50 

.64528 

1.54972 

.67028 

1.49190 

.69588 

>.  43703 

72211 

1.38484 

10 

51 

.64569 

1.54873 

.67071 

1.49097 

.69631 

1.43614 

.72255 

1.38399 

9 

52 

.64610 

1.54774 

.67113 

1.49003 

.69675 

1.43525 

.72299 

1.38314 

8 

53 

.64652 

1.54675 

!    .67155 

1.48909 

.69718 

1.43436 

.72344 

1  38229 

54 

.64693 

1.54576 

!    .67197 

1.48816 

.69761 

1.43347 

.72388 

1.38145 

6 

55 

.64734 

1.54478 

j    .67239 

1.48722 

.69804 

1.43258 

.72432 

1  38060 

g 

56 

.64775 

1.54379 

.67282 

1.48629 

.69847 

1.43169 

.72477 

1.37976 

^ 

57 

.64817 

1.54281 

.67324 

1.48536 

.69891 

1.43080 

.72521 

1  37891 

3 

58 

.64858 

1.54183 

.67366 

1.48442 

.69934 

1.42992 

.72565 

1.37807 

2 

58 

.64899 

1.54085 

.67409 

1.48349 

.69977 

1.42903 

.72610 

1.37722 

1 

6C 

.64941 

1.53986 

.67451 

1.48256       .70021 

1.42815 

.72654 

1.37638 

0 

/ 

Cotang     Tang 

Cotang 

Tang    '  Cotang  I    Tang 

Cotang 

Tang 

f 

57° 

56°                       55°                        54° 

317 


TABLE  XII.— TANGENTS  AND  COTANGENTS. 


3 

s° 

3 

7° 

3 

5° 

3 

9° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

0 

.72654 

1.37638 

.75355 

1.32704 

.78121) 

1.27994 

.80978 

1.23490 

on 

1 

.72699 

1.37554 

.75401 

1.32624  i 

.78175 

1.27917 

.81027 

1.23416 

69 

2 

.72743 

1.37470 

.75447 

1.32544 

.78222 

1.27841 

.81075 

1.23343 

58 

8 

.72788 

1.37386 

.75492 

1.32464 

.78269 

1.27764 

.81123 

1.23270 

67 

4 

.72832 

1.37302 

.75538 

1.32384 

.78316 

1.27688 

.81171 

1.23196 

56 

5 

.72877 

1.37218 

.75584 

1.32304 

.78363 

1.27611 

.81220 

1.23123 

66 

6 

.72921 

1.37134 

.75629 

1.32224 

.78410 

1.27535 

.81268 

1.23050 

64 

7 

.72966 

1.37050 

.75675 

1.32144 

.78457 

1.27458 

.81316 

1.22977 

63 

8 

.73010 

1.36967 

.75721 

.32064 

.78504 

1.27382 

.81364 

1.22904 

52 

9 

.73055 

1.36883 

.75767 

.31984 

.78551 

1.27306 

.81413 

1.22831 

51 

10 

.73100 

1.36800 

.75812 

.31904 

.78598 

1.27230 

.81461 

1.22758 

60 

11 

.73144 

1.36716 

.75858 

.31825 

.78645 

1.27153 

.81510 

1.22685 

•1!» 

18 

.73189 

1.36033 

.75904 

.31745 

.78692 

1.27077 

.81558 

1.22612 

•18 

13 

.73234 

1.36549 

.75950 

.31666 

.78739 

1.27001 

.81606 

1.22539 

47 

14 

.73278 

1.36466 

.75996 

.31586 

.78786 

1.26925 

.81655 

1.22467 

46 

15 

.73323 

1.36383 

.76042 

.31507 

.78834 

1.26849 

.81703 

1.22394 

45 

16 

.73368 

1.36300 

.76088 

.31427 

.78881 

1.26774 

.81752 

1.22321 

44 

17 

.73413 

1.36217 

.76134 

.31348 

.78928 

1.26698 

.81800 

1.22249 

43 

18 

.73457 

1.36134 

.76180 

1.31269 

.78975 

1.26622 

.81849 

1.22176 

42 

19 

.73502 

1.36051 

.76226 

1.31190 

.79022 

1.26,546 

.81898 

1.22104 

•11 

20 

.73547 

1.35968 

.76272 

1.31110 

.79070 

1.26471 

.81946 

1.22031 

40 

21 

.73592 

1.35885 

.76318 

1.31031 

.79117 

1.26395 

.81995 

1.21959 

39 

22 

.73637 

1.35802 

.76364 

1.30952 

.79164 

1.26319 

182044 

1.21886 

38 

23 

.73681 

1.35719 

.76410 

1.30873 

.79212 

1.26244 

.82092 

1.21814 

37 

24 

.73726 

1.35637 

.76456 

1.30795 

.79259 

1.26169 

.82141 

1.21742 

36 

25 

.73771 

1.35554 

.76502 

1.30716 

.79306 

1.26093 

.82190 

1.21670 

36 

26 

.73816 

1.35472 

.76548 

1.30637 

.79354 

1.26018 

.82238 

.21598 

34 

2V 

.73861 

1.35389 

.76594 

1.30558 

.79401 

1.25943 

.82287 

.21526 

33 

28 

.73906 

1.35307 

.76640 

1.30480 

.79449 

1.25867 

.82336 

.21454 

32 

29 

.73951 

1.35224 

.76686 

1.30401 

.79496 

1.25792 

.82385 

.21382 

81 

30 

.73996 

1.35142 

.76733 

1.30323 

.79544 

1.25717 

.824:34 

.21310 

30 

81 

.74041 

1.35060 

.76779 

1.30244 

.79591 

1.25642 

.82483 

.21238 

29 

32 

.74086 

1.34978 

.76825 

1.30166 

.79639 

1.25567 

.82531 

.21166 

28 

83 

.74131 

1.34896 

.76871 

1.30087 

.79686 

1.25492 

.82580 

.21094 

27 

34 

.74176 

1.34814 

.76918 

1.30009 

.79734 

1.25417 

.82629 

.21023 

26 

35 

.74221 

1.34732 

.76964 

1.29931 

.79781 

1.25343 

.82678 

.20951 

25 

86 

.74267 

1.34650 

.77010 

1.29853 

.79829 

1.25268 

84>727 

.20879 

21 

37 

.74312 

1.34568 

.77057 

.29775 

.79877 

1.25193 

!  82776 

.20808 

23 

88 

.74357 

1.34487 

.77103 

.29696 

.79924 

1.25118 

.82825 

.20736 

°2 

39 

.74402 

1.34405 

.77149 

.29618 

.79972 

1.25044 

.82874 

.20665 

21 

40 

.74447 

1.34323 

.77196 

.29541 

.80020 

1.24969 

.82923 

.20593 

20 

41 

.74492 

1.34242 

.77242 

.29463 

.80067 

1.24895 

.82972 

.20522 

19 

42 

.74538 

1.34100 

.77289 

1.29385 

.80115 

1.24820 

.83022 

.20451 

18 

43 

.74583 

1.34079 

.77335 

1.29307 

.80163 

1.24746 

.83071 

.20379 

17 

44 

.74628 

1.33998 

.77382 

1.29229 

.80211 

1.24672 

.83120 

.20308 

16 

46 

.74674 

1.33916 

.77428 

1.29152 

.80258 

1.24597 

.8:11  69 

.20237 

15 

46 

.74719 

1.33835 

.  77475 

1.29074 

.80306 

1.24523 

.83218 

.20166 

11 

47 

.74764 

1.33754 

.77521 

1.28997 

.80354 

1.24449 

.83268 

.20095 

18 

48 

.74810 

1.33673 

.77568 

1.28919 

.80402 

1.24375 

.83317 

.20024 

12 

48 

.74855 

1.33592 

.77615 

1.28842 

.80450 

1.24301 

,83366 

.19953 

11 

60 

.74900 

1.33511 

.77661 

1.28764 

.80498 

1.24227 

.83415 

.19882 

10 

51 

.74946 

1.33430 

.77708 

1.28687 

.80546 

1.24153 

.83465 

.19811 

9 

52 

.74991 

1.33349 

.77754 

1.28610 

.80594 

1.24079 

.83514 

.197'40 

8 

53 

.75037 

1.33268 

.77801 

1.28533 

.80642 

1.24005 

.83564 

.19669 

7 

54 

.75082 

1.33187 

.77848 

.28-156 

.80690 

1.23931 

.83613 

.19599 

6 

55 

.75128 

1.33107 

.77895 

.28379 

.80738 

1.23858 

.83662 

.19528 

5 

66 

.75173 

1.33026 

.77941 

.28302 

.80786 

1.23784 

.83712 

1.19457 

4 

67 

.75219 

1.32946 

.77988 

.28225 

.80834 

1.23710 

.83761 

1.19387 

3 

58 

.75264 

1.3:28(15 

.78035 

.28148 

.80882 

1.23637 

.&38H 

1.19316 

2 

59 

.75310 

1.32785 

.78082 

1.28071 

.80930 

1.23563 

i   .83860 

1.19246 

1 

60 

.75355 

1.32704 

.78129 

1.27994 

.80978 

1.23490 

.83910 

1.19175 

0 

/ 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

;  Cotang 

Tang 

. 

i 

3° 

5 

2° 

5 

1° 

!           5 

0° 

318 


TABLE  XII.-TANGENTS  AND   COTANGENTS. 


40°  .                     41°           !            42° 

43° 

Tang  1  Cotang  |    Tang 

Cotang 

Tang     Cotang 

Tang     Cotang 

0 

.S3'.)  10      1.19175 

.86929 

1.15037 

.90040     1.11061 

.93252 

1.07237 

60 

1 

.83960      1.19105 

.86980 

1.14969 

.90093 

1.10996 

.93306 

1.07174 

59 

2 

.84009      1.19035 

.87031 

.14902 

.90146 

1.10931 

.93360 

1.07112 

58 

3 

.84059 

1.18964 

.87082 

.14834 

.90199 

1.10867 

.93415 

1.07049 

57 

4 

.84108 

1.18894 

.87133 

.14767 

.90251 

1.10802 

.93469 

1.06987 

56 

6 

.84158 

1.18824 

.87184 

.14699 

.90304 

1.10737 

.93524 

1.06925 

55 

6 

.84208 

1.18754 

.87236 

.14632 

.90357 

1.10672 

.93578 

1.06862 

51 

7 

.84258 

1.18684 

.87287 

.14565 

.90410 

1.10607 

.93633 

1.06800 

53 

S 

.84307 

1.18614 

.87338  i     .14498 

.90463 

1.10543 

.93688 

1.06738 

5:3 

<) 

.84357 

1.18544 

.87389 

.14430 

.90516  i  1.10478 

.93742 

1.06676 

51 

10 

.84407 

1.18474 

.87441 

.14363 

.90569      1.10414 

.93797 

1.06613 

50 

11 

.84457 

1.18404 

.87492 

.14296 

.90621      1.10349 

.93852 

.06551 

41) 

12 

.84507 

1.18334 

.87543 

.14229 

.90674 

1.10285 

!   .93906 

.06489 

4S 

13 

.84556 

1.18264 

.87595 

.14162 

.90727 

1.10220 

.93961 

.06427 

47 

14 

.84606 

1.18194 

.87646 

.14095 

.90781      1.10156 

.94016 

.06365 

46 

15 

.84656 

1.18125 

.87698 

1.14028 

.90834     1.10091 

'    .94071 

.06303 

45 

16 

.84706 

1.18055 

.87749     1.13961    !    .90887     1.10027 

.94125 

.06241 

44 

17     .84756 

1.17986 

.87801      1.13894 

i    .90940  !  1.09963 

!   .94180 

.06179 

43 

18 

.84806 

1.17916 

.87852 

1.13828 

.90993  '  1.09899 

.94235 

.06117 

42 

19 

.84856 

1.17846 

.87904 

1.13761 

.91046      1.09834 

!   .94290 

.06056 

41 

90 

.84906 

1.17777 

.87955 

1.13694       .91099     1.09770 

.94345 

.05994 

40 

21 

.84956 

1.17708 

.88007 

1.13627       .91153     1.09706 

.94400 

.05932 

89 

22 

.85006 

1.17638 

.88059 

1.13561 

.91206 

1.09642 

.94455 

.05870 

38 

23 

.85057      1.17569 

.88110 

1.13494 

.91259 

1.09578 

.94510 

.05809 

37 

34 

.85107 

1  .  17500 

.88162 

1.13428 

.91313 

1.09514 

.94565 

.05747 

36 

25 

.85157 

1.17430 

.88214 

1.13361 

.91366 

1.09450 

i   .94620 

.05685 

85 

26 

.85207 

1.17361 

.88265 

1.13295 

.91419 

1.09386 

.94676 

.05624 

84 

27 

.85257 

1.17292 

.88317 

1.13223 

.91473 

1.09322 

.94731 

.05562 

33 

28     .85308 

1.17223 

.88369 

1.13162 

.91526 

1.09258 

.94786 

.05501 

32 

291    .85358 

1.17154 

.88421 

1.13096 

.91580 

1.09195 

.94841 

.05439 

31 

30 

.85408 

1.17085 

.88473 

1.13029 

.91633 

1.09131 

.94896 

.05378 

30 

81 

.85458 

1.17016 

.88524 

1.12963 

.91687 

1.09067 

.94952 

.05317 

29 

32 

.85509 

1.16947 

.88576 

1.12897 

.91740 

1.09003 

.95007 

.05255 

28 

83 

.85559 

1.16878 

.88628 

1.12831 

.91794 

1.08940 

.95062 

.05194 

27 

34 

.85609 

1.16809 

.88680 

1.12765 

.91847 

1..  08876 

.95118 

.05133 

26 

85 

.85660 

1.16741 

.88732 

1.12699 

.91901 

1.08813 

.95173 

.05072 

25 

36 

.85710 

1.16672 

.88784 

1.12633 

.91955      1.08749 

.95229 

.05010 

21 

37 

.85761 

1.16603 

.88836 

1.12567 

.92008     1.08686 

.95284 

.04949 

23 

38 

.85811 

1.16535 

.88888 

1.12501 

.92062  1  1.08622 

.95340 

1.04888 

22 

39 

.85862 

1.16466 

.88940 

1.12435 

.92116     1.08559 

.95395 

1.04827 

21 

40 

.85912 

1.16398 

.88992 

1.12369 

.92170 

1.08496 

.95451 

1.04766 

20 

41 

.85963 

1.16329 

.89045 

1.12303 

.92224 

1.08432 

.95506 

1.04705 

19 

42 

.86014 

1.16261 

.89097 

1.12238 

.92277 

1.08369 

.95562 

1.04644 

18 

43 

.86064 

1.16192 

.89149 

1.12172 

.92331 

1.08306 

.95618 

1.04583 

17 

44 

.86115 

1.16124 

.89201 

1.12106 

.92385 

1.08243 

.95673 

1.04522 

16 

45 

.86166 

1.16056 

.89253 

1.12041 

.92439  i  1.08179 

.95729 

1.04461 

15 

46 

.86216 

1.15987 

.89306 

1.11975 

.92493 

1.08116 

.95785 

1.04401 

14 

47 

.86267 

1.15919 

.89358 

1.11909 

.92547 

1.08053 

.95841 

1.04340 

13 

48 

.86318 

1.15851 

.89410 

1.11844 

.92601 

1-07990 

.95897 

1.04279 

12 

491    .86368 

1  .  15783 

.89463 

1  11778 

.92655 

1.07927 

.95952 

1.04218 

11 

50 

.86419 

1.15715 

.89515 

1.11713 

.92709 

,1.07864 

.96008 

1.04158 

10 

51 

.86470 

1.15647 

.89567 

1.11648 

.92763 

1.07801 

.96064 

1.04097 

9 

52!    .86521 

1  .  15579 

.89620 

1.11582 

.92817 

1.07738 

.96120 

1.04036 

8 

53 

.86572 

1.15511 

.8967'2 

1.11517 

.92872 

1.07676 

.96176 

1.03976 

7 

54 

.86623 

1.15443 

.89725 

1.11452 

.92926 

1.07613 

.96232 

1.03915 

6 

55     .86674 

1.15375 

.89777 

1.11387 

.92980 

1.07550 

.96288 

1.03855 

5 

56 

.86725 

1.15308 

..89&30 

1.11321 

.93034 

1.07487 

.96344 

1.03794 

4 

57 

.86776 

1.15240 

.89883 

1.11256 

.93088 

1.07425 

.96400 

1.03734 

3 

58 

.86827 

1.15172 

.89935 

1.11191 

.93143 

1.07362 

.96457 

1.03674 

2 

59 

.86878 

1.15104 

.89988 

1.11126 

i   .93197 

1.07299 

.96513 

1.03613 

1 

60 

.86929 

1.15037 

.90040     1.11061  J|   .93252 

1.07237 

.96569 

1.03553 

0 

f 

Cotang     Tang      Cotang     Tang      Cotang 

Tang 

jCotang     Tang 

49°           !l           48°           i            47°           li           46° 

319 


TABLE  XII.-TANGENTS  AND  COTANGENTS. 


f 

4 

t4° 

4 

t4° 

4 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

0 

.96569 

1.08553 

60 

20 

.97700 

1.02355 

41) 

40 

.98843 

1.01170 

20 

1 

.96625 

1.03493 

59 

21 

.97756 

1.02295 

39 

141 

.98901 

1.01112 

19 

2 

.96681 

1.03433 

58 

22 

.97813 

1.02236 

as 

I42 

.98958 

1.01053 

18 

3 

.96738 

1.03372 

57 

23 

.97870 

1.02176 

37 

43 

.99016 

1.00994 

17 

4 

.96794 

1.03312 

56 

24 

.97927 

1.02117 

36 

44 

.99073 

1.00935 

16 

5 

.96850 

1.03252 

55 

25 

.97984 

1.02057 

35 

45 

.99131 

1.00876 

15 

6 

.96907 

1.03192 

54 

26 

.98041 

1.01998 

34 

4(5 

.99189 

1.  d0818 

14 

7 

.96963 

1.03132 

53 

27 

.98098 

1.01939 

33 

!47 

.99247 

1.00759 

13 

8 

.97020 

1.03072 

53 

28 

.98155 

1.01879 

32 

48 

.99304 

1.00701 

12 

9 

.97076 

1.03012 

51 

29 

.98213 

1.01820 

31 

49 

.99362 

1.00642 

11 

10 

.971&3 

1.02952 

50 

1  30 

.98270 

1.01761 

30 

J50 

.99420 

1.00583 

10 

11 

.97189 

1.02892 

49 

81 

.98327 

.01702 

29 

51 

.99478 

1.00525 

9 

12 

.97246 

1.02832 

48 

32 

.98384 

.01642 

28 

52 

.99536 

1.00467 

8 

13 

.97302 

1.02772 

47 

38 

.98441 

.01583 

27 

53 

.99594 

1.00408 

7 

14 

.97359 

1.02713 

46 

34 

.98499 

.01524 

26 

54 

.99652 

1.00350 

6 

15 

.97416 

1.02653 

45 

35 

.98556 

.01465 

25 

55 

.99710 

1.00291 

5 

10 

.97472 

1.02593 

44 

36 

.98613 

.01406 

24 

56 

.99768 

1.00233 

4 

1? 

.97529 

1.02533 

43 

37 

.98671 

.01347 

23 

57 

.99826 

1.00175 

3 

18 

.97586 

1.02474 

42 

38 

.98728 

.01288 

22 

58 

.99884 

1.00116 

a 

19 

.97643 

1.02414 

41 

39 

.98786 

.01229 

21 

59 

.99942 

1.00058 

i 

20 

.97700 

1.02355 

40 

40 

.98843 

1.01170 

20 

60 

1.00000 

1.00000 

0 

Cotang 

Tang 

i 

Cotang 

Tang 

f  j 

/ 

Cotang 

Tang 

/ 

4 

5° 

4 

5° 

4 

5° 

320 


TABLE  XIII.-VERSINES  AND  EXSECANTS. 


0\ 

; 
1° 

2° 

3° 

/ 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.00000   .00000 

.00015 

.00015 

.00061 

.00061 

.00137 

.00137 

0 

1 

.00000   .00000 

.00016 

.00016 

.00062 

.00062 

.00139 

00139 

1 

2 

.00000   .00000 

.00016 

.00016 

.00063 

.00063 

.00140 

.00140 

2 

3 

.00000  !  .00000 

.00017 

.00017 

.00064 

.00064  i 

.00142 

.00142 

3 

4 

.00000 

.00000 

.00017 

.00017 

.00065 

.00065 

.00143 

.00143 

.4 

5 

.00000 

.00000 

[  .00018 

.00018 

.00066 

.00066 

.00145 

.00145 

5 

6 

.00000 

.00000 

.00018 

.00018 

.00067 

.00067  I 

.00146 

.00147 

6 

y 

.00000 

.00000 

.00019 

.00019 

.00068 

.00068  i 

.00148 

.00148 

7 

8 

.00000 

.00000 

!  .00020 

.00020 

.00069 

.00069  i 

.00150 

.00150 

8 

9 

.00000 

.00000 

i  .00020 

.00020 

.00070 

.00070 

.00151 

.00151 

9 

10 

.00000 

.00000 

j  .00021 

.00021 

.00071 

.00072 

.00153 

.00153 

10 

11 

.00001 

.00001 

.00021 

.00021 

.00073 

.00073 

.00154 

.00155 

11 

12 

.00001 

.00001 

!  .00022 

.00022 

.00074 

.00074 

.00156 

.00156 

12 

13 

.00001 

.00001 

.00023 

.00023 

.00075 

.00075 

.00158 

.00158 

13 

14 

.00001 

.00001 

.00023 

.00023 

.00076 

.00076 

.00159 

.00159 

14 

15 

.00001 

.00001 

.00024 

.00024 

.00077 

.00077 

.00161 

.00161 

15 

16 

.00001 

.00001 

:  .00024 

.00024 

.00078 

.00078 

.00162 

.00163 

16 

17 

.00001 

.00001 

.00025 

.00025 

.00079 

.00079 

.00164 

.00164 

17 

18 

.00001 

.00001 

.00026 

.00026 

.00081 

.00081 

.00166 

.00166 

18 

19 

.00002 

.00002 

i  .00026 

.00026 

.00082 

.00082 

.00168 

.00168 

19 

20 

.00002 

.00002 

.00027 

.00027 

.00083 

.00083 

.00169 

.00169 

20 

21 

.00002 

.00002 

.00028 

.00028 

.00084 

.09084 

.00171 

.00171 

21 

22 

.00002 

.00002 

.00028 

.00028 

.00085 

.00085 

.00173 

.00173 

22 

23 

.00002 

.00002 

.00029 

.00029 

.00087 

.00087 

.00174 

.00175 

23 

24 

.00002 

.00002 

.00030 

.00030 

.00088 

.00088 

.00176 

.00176 

24 

25 

.00003 

.00003 

.00031 

.00031 

.00089 

.00089 

.00178 

.00178 

25 

26 

.00003 

.00003 

.00031 

.00031 

.00090 

.00090 

.00179 

.00180 

26 

27 

.00003 

.00003 

.00032 

.00032 

.00091 

.00091 

.00181 

.00182 

27 

28 

.00003 

.00003 

.00033 

.00033 

.00093 

.00093 

.00183 

.00183 

28 

29 

.00004 

.00004 

.00034 

.00034 

.00094 

.00094 

.00185 

.00185 

29 

30 

.00004 

.00004 

.00034 

.00034 

.00095 

.00095 

.00187 

.00187 

30 

31 

.00004 

.00004 

.00035 

.00035 

.00096 

.00097 

.00188 

.00189 

31 

32 

.00004 

.00004 

.00036 

.00036 

.00098 

.00098 

.00190 

.00190 

32 

33 

.00005 

.00005. 

.00037 

.00037 

.00099 

.00099 

.00192 

.00192 

33 

34 

.00005 

.00005 

.00037 

.00037 

.00100 

.00100 

.00194 

.00194 

34 

35   .00005 

.00005 

.000:38 

.00038 

.00102 

.00102 

.00196 

.00196 

35 

30  :  .00005 

.00005 

.00039 

.00039 

.00103 

.00103 

.00197 

.00198 

36 

37 

.00006 

.00006 

1  .00040 

.00040 

.00104 

.00104 

.00199 

.00200 

37 

38 

.00006 

.00006 

i  .00041 

.00041 

.00106 

.00106 

.00201 

.00201 

38 

39 

.00006 

.00006 

.00041 

.00041 

.00107 

.00107 

.00203 

.00203 

39 

40 

.00007 

.00007 

i  .00042 

.00042 

.00108 

.00108 

.00205 

.00205 

40 

41 

.00007 

.00007 

.00043 

.00043 

.00110 

.00110 

.00207 

.00207 

41 

42 

.00007 

.00007 

.00044 

.00044 

.00111 

.00111 

.00208 

.00209 

42 

43 

.00008 

.00008 

.00045 

.00045 

.00112 

.00113 

.00210 

.00211 

43 

44 

.00008 

.00008 

.00046 

.00046 

.00114 

.00114 

.00212 

.00213 

44 

45 

.00009 

.00009 

.00047 

.00047 

.00115 

.00115 

.00214 

.00215 

45 

46 

.00009 

.00009 

.00048 

.00048 

.00117 

.00117 

.00216 

.00216 

46 

47 

.00009 

.00009 

.00048 

.00048 

.00118 

r00118 

.00218 

.00218 

47 

48 

.00010 

.00010 

.00049 

.00049 

.00119 

.00120 

.00220 

.00220 

48 

49 

.00010  !  .00010 

.00050 

.00050 

.00121 

.00121  - 

.00222 

.00222 

49 

50 

.00011 

.00011 

.00051 

.00051 

.00122 

.00122 

.00224 

.00224 

50 

51 

.00011 

.00011 

.00052 

.00052 

.00124 

.00124 

.00226 

.00226 

51 

52 

.00011 

.00011 

.00053 

.00053 

.00125 

.00125 

.00228 

.00228 

52 

53 

.00012 

.00012 

-.00054 

.00054 

.00127 

.00127 

.00230 

.00230 

53 

54 

.00012 

.00012  1  .00055 

.00055 

.00128 

.00128 

.00232 

.00232 

54 

55 

.00013 

.00013 

.00056 

.00056 

.00130 

.00130 

.00234 

.00234 

55 

56 

.00013 

.00013 

.00057 

.00057 

.00131 

.00131 

.00236 

.00236 

56 

57 

.00014 

.00014 

.00058 

.00058 

.00133 

.00133 

.00238 

.00238 

57 

58 

.00014 

.00014 

j  .00059 

.00059 

.00134 

.00134  ! 

.00240 

.00240 

58 

59 

.00015 

.00015 

.00060 

.00060  i 

.00136 

.00136 

.00242 

.00242 

59 

60 

.00015 

.00015 

.00061 

.00061  i 

.00137 

.00137 

.00244 

.00244 

60 

321 


TABLE  XIII.— VERSINES  AND  EXSECANTS. 


• 

4° 

5C 

6° 

7° 

' 

Vers. 

Exsec. 

Vers.- 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.00244 

.00244 

.00:381 

.00382 

.00548 

.00551 

.00745 

.00751   0 

1 

.00246 

.00246 

.00383 

.00385 

.00551 

.00554   .00749 

.00755  !  1 

2 

.00248 

.00248 

.00386 

.00387 

.00554 

.00557   .00752   .(.0758   2 

3 

.00250 

.00250 

.00388 

.00390 

.00557 

.00560   .00756   .00762 

3 

4 

.00252 

.00252 

.00391 

.00392 

.00560 

.00563 

.00760   .00765 

4 

5 

.00254 

.00254 

.00393 

.00395 

.00563 

.00566  i 

.00763   .00769   5 

6 

.00256 

.00257 

.00396 

.00397 

.00566 

.00569 

.00767 

.00773  !  6 

7 

.00258 

.00259 

.00398 

.00400 

.00569 

.00573 

.00770 

.00776   7 

8 

.00260 

.00261 

.00401 

.00403 

.00572 

.00576 

.00774 

.00780 

8 

9 

.00262 

.00263 

.00404 

.00405 

.00576 

.00579 

.00778 

.00784 

9 

10 

.00264 

.00265 

.00406 

.00408 

.00579 

.00582 

.00781 

.00787 

10 

11 

.00266 

.00267 

.00409 

.00411 

.00582 

.00585  '• 

.00785 

.00791 

11 

12 

.00269 

.00269 

.00412 

.00413 

.00585 

.00588 

.00789 

.00795 

12 

13 

.00271 

.00271 

.00414 

.00416 

.00588 

.00592 

.00792 

.00799 

13 

14 

.00273 

.00274 

.00417 

.00419 

.00591 

.00595  ! 

.00796 

.00802 

14 

15 

.00275 

.00276 

.00420 

.00421 

.00594 

..00598  ! 

.00800 

.00806 

15 

16 

.00277 

.00278 

.00422 

,00424 

.00598 

.00601 

.00803 

.00810 

16 

17 

.00279 

.00280 

.00425 

.00427 

.00601 

.00604 

.00807 

.00813 

17 

18 

.00281 

.00282 

.00428 

.00429 

.00604 

.00608  j 

.00811 

.00817 

18 

19 

.00284 

.00284 

.00430 

.00432 

.00607 

.00611 

.00814 

.00821 

19 

20 

.00286 

.00287 

.00433 

.00435 

.00610 

.00614 

.00818 

.00825 

20 

21 

.00288 

.00289 

.00436 

.00438 

.00614 

.00617 

.00822 

.00828 

21 

22 

.00290 

.00291 

.00438 

.00440 

.00617 

.00621 

.00825 

.00832 

22 

23 

.00293 

.00293 

.00441 

.00443 

.00620 

.00624 

.00829 

.00836 

23 

24 

.00295 

.00296 

.00444 

.00446 

.00623 

.00627  i 

.00833   .00840 

24 

25 

.00297 

.00298 

.00447 

.00449 

.00626 

.00630 

.00837   .00844 

25 

26 

.00299 

.00300 

.00449 

.00451 

.00630 

.00634 

.00840  •  .00848 

26 

27 

.00301 

.00302 

.00452 

.00454 

!  00633 

.00637 

.00844  ;  .00851 

27 

28 

.00304 

.00305 

.00455 

.00457 

.00636 

.00640 

.00848   .00855  |  28 

29 

.00306 

.00307 

.00458 

.00460 

.00640 

.00644 

.00852   .00859 

29 

30 

.00308 

.00309 

.00460 

.00463 

.00643 

.00647 

.00856 

.00863 

30 

31 

.00311 

.00312 

.00463 

.00465 

.00646 

.00650 

.00859 

.00867 

31 

32 

.00353 

.00314 

.00466 

.00468 

.00649 

.00654  i 

.00863 

.00871 

32 

33 

.00315 

.00316 

.00469 

.00471 

.00653 

.00657 

..00867  !  .00875 

33 

34 

.00317 

.00318 

.00472 

.00474 

.00656 

.00660 

.00871  !  .00878 

34 

35 

.00320 

.00321 

.00474 

.00477 

.00659 

.00664 

.00875  i  .00882 

35 

36 

.00322 

.00323 

.00477 

.00480 

.00663 

.00667 

0087'8  ;  .00886 

36 

37 

.00324 

.00326 

.00480 

.00482 

.00666 

.00671  j 

.00882  :  .00890 

37 

38 

.00327 

.00328 

.00483 

.00485 

.00669 

.00674 

.00886   .00894 

38 

39 

.00329 

.00330 

.00486 

.00488 

.00673 

.00677 

00890   .00898 

39 

40 

.00333 

.00489 

.00491 

.00676 

.00681 

.00894 

.00902 

40 

41 

.00334 

.00335 

.00492 

.00494 

.00680 

.00684 

.00898 

.00906 

41 

42 

.00336 

.00337 

.00494 

.00497 

.00683 

.00688 

00902   .00010 

42 

43 

.00339 

.00340 

.00497 

.00500 

.00686 

.00691 

.00906  !  .00914 

43 

44 

.00341 

.00342 

.00500 

.00503 

.00690 

.00695 

.00901)   .00918 

44 

45 

.00:343 

.00345 

.00503 

.00506 

.00693 

.00698 

.00913   .00922 

45 

46 

.00346 

.00347 

.00506 

.00509 

.00697 

.00701 

.00917  '  .00926 

46 

47 

.00348 

.00350 

.00509 

.00512 

.00700 

.00705 

.00921  :  .00930 

47 

48 

.00351 

.00352 

.00512 

.00515 

.00703 

.00708 

.000% 

.001134 

48 

49 

.00353 

.00354 

.00515 

.00518 

.00707 

.00712 

.00929 

.00938 

49 

50 

.00356 

.00357 

.00518 

.  00081 

.00710 

.00715 

.00933 

.00942 

50 

51 

00358 

.00359 

.00521 

.008*4 

.00714 

.00719 

.00937 

00946 

51 

52 

!  00361 

.00362 

.00524 

.00527 

.00717 

.00722 

.00941 

.00960 

52 

53 

.00363 

.00364 

.00527 

.00530 

.00721 

.00726 

.00945 

.00954 

53 

54 

.00365 

.00367 

.00530 

.00533 

.00724 

.00730 

.00949 

.00958 

54 

55 

.00368 

.00369 

.00533 

.00536 

.00728 

.00733 

.00953 

.00962 

55 

56 

.00370 

.00372 

.00536 

.00539 

.00731 

.00737 

.00957 

.00966 

56 

57 

.00373 

.00374 

.00539 

.00542 

.00735 

.00740 

.009(11 

.00970 

57 

58 

.00375 

.00377 

.00542 

.00545 

.00738 

.00744 

.00965 

.00975 

58 

59 

.00378 

.00379 

.00545 

.00548 

.00742 

.00747 

.00969 

.00979 

59 

60 

.00381 

.00382 

.00548 

.00551 

.00745 

.00751 

.00973 

.00983 

60 

823 


TABLE  XIII.-VERSINES  AND  EXSECANTS. 


/ 

8° 

9° 

10° 

11° 

/ 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.00973 

.00983 

.01231 

.01247 

.01519 

.01543   .01837 

.01872 

0 

1 

.00977 

.00987 

.01236 

.01251  I  .01524 

.01548  I 

.01843  1  .01877 

1 

2 

.00981 

.00991 

.01240 

.01256 

.01529 

.01553 

.01848 

.01883 

2 

3 

.00985 

.00995 

.01245 

.01261 

.01534 

.01558 

.01854 

.01889 

3 

4 

.00989 

.00999 

.01249 

.01265 

.01540 

.01564 

.01860 

.01895 

4 

5 

.00994 

.01004 

.01254 

.01270 

.01545 

.01569 

.01865 

.01901 

5 

6 

.00998 

.01008  ! 

.01259 

.01275 

.01550 

.01574 

.01871 

.01906 

6 

7 

.01002 

.01012 

.01263 

.01279 

.01555 

.01579 

.01876 

.01912 

7 

8   .01006 

.01016  ! 

.01268 

.01284 

.01560 

.01585 

.01882 

.01918 

8 

9 

.01010 

.01020 

.01272 

.01289 

.01565 

.01590 

.01888 

.01924 

9 

10 

.01014 

.01024 

.01277 

.01294 

.01570 

.01595 

.01893 

.01930 

10 

11 

.01018 

.01029 

.01282 

.01298 

.01575 

.01601  1 

.01899 

.01936 

11 

13 

.01022 

.01033  ! 

.01286 

.01303   .01580 

.01606 

.01904 

.01941 

12 

13 

.01027 

.01037 

.01291 

.01308 

.01586 

.01611 

.01910 

.01947 

13 

14 

.01031 

.01041 

.01296 

.01313 

.01591 

.01616 

.01916 

.01953 

14 

15 

.01035 

.01046 

.01300 

.01318 

.01596 

.01622  ! 

.01921 

.01959 

15 

16 

.01039 

.01050 

.01305 

.01322 

.01601 

.01627 

.01927 

.01965 

16 

17 

.01043 

.01054 

.01310 

.01327 

.01606 

.01633 

.01933 

.01971 

17 

18 

.01047 

.01059 

.01314 

.01332 

.01612 

.01638 

.01939 

.01977 

18 

19 

.01052 

.01063  i 

.01319 

.01.337 

.01617 

.01643 

.01944 

.01983 

19 

20 

.01056 

.01067 

.01324 

.01342 

.01622 

.01649 

.01950 

.01989 

20 

21 

.01060 

.01071 

.01329 

.01346 

.01627 

.01654 

.01956 

.01995 

21 

22 

.01064 

.01076 

.01333 

.01351 

.01032 

.01659 

.01961 

.02001 

22 

28 

.01069 

.01080 

.01338 

.01356 

.01638 

.01665 

.01967 

.02007 

23 

24 

.01073 

.01084 

.01343 

.01361 

.01643 

.01670 

.01973 

.02013 

24 

23 

.01077 

.01089 

.01348 

.01366 

.01648 

.01676 

.01979 

.02019 

25 

23 

.01081 

.01093 

.01352 

.01371 

.01653 

.01681 

.01984 

.02025 

26 

27 

.01086 

.01097  : 

.01357 

.01376 

.01659 

.01687 

.01990 

.02031 

27 

28 

.01090 

.01102  i 

.01362 

.01381 

.01664 

.01692 

.01996 

.02037 

28 

29 

.01094 

.01106  ! 

.01367 

.01386 

.01669 

.01698 

.02002 

.02043 

29 

30 

.01098 

.01111 

.01371 

.01391 

.01675 

.01703 

.02008 

.02049 

30 

31 

.01103 

.01115 

.01376 

.01395 

.01680 

.01709 

.02013 

.02055 

31 

32 

.01107 

.01119 

.01381 

.01400 

.01685 

.01714 

.02019 

.02061 

32 

33 

.01111 

.01124  : 

.01386 

.01405 

.01690 

.01720 

.02025 

.02067 

33 

34 

.01116 

.01128  j 

.01891 

.01410 

.01696 

.01725 

.02031 

.02073 

34 

35 

.01120 

.01133 

.01396 

.01415 

.01701 

.01731 

.02037 

.02079 

35 

36 

.01124 

.01137  i 

.01400 

.01420 

.01706 

.0-1736 

.02042 

.02085 

36 

37 

.01129 

.01142  ! 

.01405 

.01425 

.01712 

.01742 

.02048 

.02091 

37 

38 

.01133 

.01146  j 

.01410 

.01-130 

.01717 

.01747 

.02054 

.0.2097 

38 

33 

.01137 

.01151  ! 

.01415 

.01435 

.01723 

.01753 

.02000 

.02103 

39 

40 

.01142 

.01155 

.01420 

.01440 

.01728 

.01758 

.02066 

.02110 

40 

41 

.01146 

.01160 

.01425 

.01445 

.01733 

.01764 

.02072 

.02116 

41 

42 

.01151 

.01164  : 

.01430 

.01450 

.01739 

.01769 

.02078 

.02122 

42 

43 

.01155 

.01169 

.01435 

.01455 

.01744 

.01775 

.02084 

.02128 

43 

44 

.01159 

.01173 

.01439 

.01461 

.01750 

.01781 

.02090 

.02134 

44 

45 

.01164 

.01178 

.01444 

.01466 

.01755 

.01786 

.02095 

.02140 

45 

46 

.01168   .01182 

.01449 

.01471 

.01760 

-  .01792 

.02101 

.02146 

46 

47 

.01173 

.01187 

.01451 

•01476 

.01766 

.01798 

.02107 

.02153 

47 

48 

.01177 

.01191 

.01459 

.01481 

.01771 

.01803 

.02113 

.02159 

'48 

49 

.01182 

.01196 

.01464 

.01486 

.01777 

.01809 

.02119 

.02165 

49 

50 

.01186 

.01200 

.01469 

.01491 

.01782 

.01815 

.02125 

.02171 

50 

51 

.01191 

.01205 

.01474 

.01496 

.01788 

.01820 

.02131 

.02178 

51 

52 

01195 

.01209 

.01479 

.01501 

.01793 

.01826 

.02137 

.02184 

52 

53 

.01200 

.01214 

.01484 

.01506 

.01799 

.01832 

.02143 

.02190 

53 

54 

.01204 

.01219 

.01489 

.01512 

.01804 

.01837  , 

.02149 

.02196 

54 

55 

.01209 

.01223 

.01494 

.01517 

.01810 

.01843  | 

.02155   .02203 

55 

56 

.01213 

.01228 

.01499 

.01522 

.01815 

.01849  : 

.02161 

.02209 

56 

57 

.01218 

.01233 

.01504 

.01527 

.01821 

.01854 

.02167 

.02215  '  57 

58 

.01222 

.01237 

.01509 

.01532 

.01826 

.01860 

.02173 

.02221   58 

59 

.01227 

.01242  ! 

.01514 

.01537 

.01832 

.01866 

.02179 

.02228 

59 

60 

.01231 

.01247 

.01519 

.01543 

.01837 

.01872  1 

.02ia5 

.02234 

60 

323 


TABLE  XIII.-VERSINES  AND  EXSECANTS. 


1 

12° 

13° 

14° 

15° 

1 

I 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.02185 

.02234 

.02563 

.02630 

.02970 

.03061 

.03407 

.03528 

0 

1 

.02191 

.02240 

.02570 

.02637 

.02977 

.03069 

.03-115 

.03536 

1 

2 

.02197 

.02247 

.02576 

.02644 

.02985 

.03076 

.03422 

.03544 

2 

3 

.02203 

.02253 

.02583 

.02651 

.02992 

.03084 

.03430 

.03552 

3 

4 

.02210 

.02259 

.02589 

.02658 

.02999 

.03091 

.03438 

.03560 

4 

5 

.02216 

.02206 

.02596 

.02665 

.03006  j  .03099 

.03445 

.03568 

5 

6 

.02222 

.02272 

.02602 

.02672 

.03013   .03106   .03453 

.03576 

6 

7 

.02228 

.02279 

.02609 

.0267'9 

.03020 

.03114   .03460 

.03584 

7* 

8 

.02234 

.02285 

.02616 

.C2686 

.03027 

.03121 

.03468 

.0:3592 

8 

9 

.02240 

.02291 

.02622 

.02093 

.03034 

.03129 

.03176 

.03601 

9 

10 

.02246 

.02298 

.02629 

.02700 

.03041 

.03137 

.03483 

.03609 

10 

11 

.02252 

.02304 

.02635 

.02707 

.08048 

.03144 

.03491 

.03617 

11 

12 

.02258 

.02311 

.02642 

.02714 

.03055 

.03152 

.03498 

.03625 

12 

13 

.02265 

.02317 

.02649 

.02721 

.03063 

.03159 

.03506 

.03633 

13 

14 

.02271 

.02323 

.02655 

.02728 

.03070 

.03167 

.03514 

.03642 

14 

15 

.02277 

.02330 

.02602 

.02735 

.03077 

.03175 

.03521 

.03650 

15 

16 

.02283 

.02336 

.02669 

.02742 

.03084 

.03182 

.03529 

.03658 

16 

17 

.02289 

.02343 

.02675 

.02749 

.03091 

.03190 

.03537 

.03666 

17 

18 

.02295 

.02349 

.02682 

.02756 

.03098 

.03198 

.03544 

.03674 

18 

19 

.02302 

.02356 

.02689 

.02763 

.03106 

.03205 

.03552 

.03683 

19 

20 

.02308 

.02362 

.02696 

.02770 

.03113 

.03213 

.03560 

.03691 

20 

21 

.02314 

.02369 

.02702 

.02777 

.03120 

.03221 

.03567 

.03699 

21 

22 

.02320 

.02375 

.02709 

.02784 

.03127 

.03228 

.03575 

.037'08 

22 

23 

.02327 

.02382 

.02716 

.02791 

.03134 

.03236 

.03583 

.03716 

23 

24 

.02333 

.02388 

.02722 

.02799 

.03142 

.03244 

.03590 

.03724 

24 

25 

.02339 

.02395 

.02729 

.02806 

.03149 

.03251 

.03598 

.03732 

25 

26 

.02345 

.02402 

.02736 

.02813 

.03156 

.03259 

.03606 

.03741 

26 

27 

.02352 

.02408 

.02743 

.02820 

.03163 

.03267 

.03614 

.03719 

27 

28 

.02358 

.02415 

.02749 

.02827 

.03171 

.03275 

.03621 

.03758 

23 

29 

.02364 

.02421 

.02756 

.02834 

.03178 

.03282 

.03629 

.03766 

29 

30 

.02370 

.02428 

.02763 

.02842 

.03185 

.03290 

.03637 

.03774 

30 

31 

.02377 

.02435 

.02770 

.028-19 

.03193 

.03298 

.03645 

.03783 

31 

32 

.02383 

.02441 

.02777 

.02856 

.03200 

.03306 

.03653 

.03791 

32 

33 

.02389 

.02448 

.02783 

.02863 

.03207 

.03313 

.03660 

.03799 

33 

34 

.02396 

.02454 

.02790 

.02870 

.03214 

.03321 

.03668 

.03808 

34 

35 

.02402 

.02461 

.02797 

.02878 

.03222 

.03329 

.03676 

.03816 

35 

36 

.02408 

.02468 

.02804 

.02885 

.03229 

.03337 

.03684 

.03825 

36 

37 

.02415 

.02474 

.02811 

.02892 

.03236 

.03345 

.03692 

.03833 

37 

38 

.02421 

.02481 

.02818 

.02899 

.03244 

.03353 

.03699 

.03842 

38 

39 

.02427 

.02488 

.02824 

.02907 

.03251 

.03360 

.03707 

.03850 

39 

40 

.02434 

.02494 

.02831 

.02914 

.03258 

.03368 

.03715 

.03858 

40 

41 

.02440 

.02501 

.02838 

.02921 

.03266 

.03376 

.03723 

.03867 

41 

42 

.02447 

.02508 

.02845 

.02928 

.03273 

.03384 

.03731 

.03875 

42 

43 

.02453 

.02515 

.02852 

.02936 

.03281 

.03392 

.03739 

.03884 

43 

44 

.02459 

.02521 

.02859 

.02943 

.03288 

.03400 

.03747 

.03892 

44 

45 

.02466 

.02528 

.02866 

.02950 

.03295 

.03408 

.03754 

.03901 

45 

46 

.02472 

.02535 

.02873 

.02958 

.03303 

.03416 

.03762 

.03909 

46 

47 

.02479 

.02542 

.02880 

.02965 

.03310 

.03421 

.03770 

.03918 

47 

48 

.02485 

.02548 

.02887 

.02972 

.03318 

.03432 

.0377'8 

.03927 

48 

49 

.02492 

.02555 

.02894 

.02980 

.03325 

.03439 

.03786 

.08835 

49 

50 

.02498 

.02562 

.02900 

.02987 

.03333 

.03447 

.03794 

.03944 

50 

51 

.02504 

.02569 

.02907 

.02994 

.03340 

.03455 

.03802 

.03952 

51 

52 

.02511 

.02576 

.02914 

.03002 

.03347 

.03463 

.03810 

.03961 

52 

53 

.02517 

.02582 

.02921 

.03009 

.03355 

.03471 

.03818 

.03969 

53 

54 

.02524 

.02589 

.02928 

.03017 

.03362 

.03479 

.03826 

.03978 

54 

55 

.02530 

.02596 

.02935 

.03024 

.03370 

.03487  ; 

.03834 

.o:',nsr  ;  55 

56 

.02537 

.02603 

.02942 

.03032 

.03377 

.0341)5 

.03842 

.03995  56 

57 

.02543 

.02C.10   .02949 

.03039 

.03385 

.03503 

.03850 

.04004 

57 

'  58 

.02550 

.02617  i  .02456 

.03046 

.03392 

.03512 

.03858 

.04013 

58 

59 

.02556 

.02624  I  .02JM53 

.03054 

.03400 

.03520 

.038(515 

.04021 

59 

60 

.02563 

.02630  II  .02970 

.03061 

.03407 

.03528 

.03874 

.04030 

60 

324 


TABLE  XIII.-VERSINES  AND  EXSECANTS. 


t 

16°         17°         18° 

19° 

/ 

Vers. 

Exsec. 

Vers. 

Exsec.   Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.03874 

.04030   .04370 

.04569  i  .04894 

.05146 

.05448  i  .05762 

0 

1 

.03882 

.04039 

.04378 

.04578   .04903 

.05156 

.05458  ;  .05773 

1 

2 

.03890 

.04047 

.04387 

.04588  '!  .04912 

.05166 

.05467  i  .05783 

2 

3 

.03898 

.04056   .04395 

.04597  i  .04921 

.05176 

.05477   .05794 

3 

4 

.03906 

.04065  i  .04404 

.04606  \  .04930 

.05186 

.05486  i  .05805 

4 

5 

.03914 

.04073   .04412 

.04616   .04939 

.05196 

.05496  :  .05815 

5 

6 

.03922 

.04082   .04421 

.04625  '  .04948 

.05206 

.05505  :  .05826 

6 

7 

.03930 

.04091   .04429 

.04635   .04957 

.05216 

.05515  i  .05836 

7 

8 

.03938 

.04100  !  .04438 

.04644   .04967 

.05226 

.05524  i  .05847 

8 

9 

.03946 

.04108  I  .04446 

.04653  |  .04976 

.05236 

.05534  !  .05858 

9 

10 

.03954 

.04117 

.04455 

.04663  ij  .04985 

.05246 

.05543 

.05869 

10 

11 

.03963 

.04120 

.04464 

.04672 

.04994 

.05256 

.05553 

.05879 

11 

12 

.03971 

.04135   .04472 

.04682 

.05003 

.05266 

.05562   .05890 

12 

13 

.03979 

.04144 

.04481 

.04691 

.05012 

.05276 

.05572   .05901 

13 

14 

.03987 

.04152 

.04489 

.04700 

.05021 

.05286 

.05582   .05911 

14 

15 

.03995 

.04161 

.04493 

.04710   .05030 

.05297   .05591   .05922 

15 

10 

.04003 

.04170 

.04507 

.04719   .05039 

.05307   .05601   .05933 

16 

17 

.04011 

.04179  : 

.04515 

.04729 

.05048 

.05317 

.05610  !  .05944 

17 

18 

.04019 

.04188  ! 

.04524 

.04738  , 

.05057 

.05327 

.05620   .05955 

18 

19 

.04028 

.04197 

.04533 

.04748 

.050G7 

.05a37  , 

.05630  (  .05965 

19 

20 

.04036 

.04206 

.04541 

.04757 

.05076 

.05347 

.05639 

.05976 

20 

21 

.04044 

.04214 

.04550 

.04767  ; 

.05085 

.05357 

.05649 

.05987 

21 

22 

.04052 

.04223  ! 

.04559 

.04776  :  .05094 

.05367 

.05658 

.05998 

22 

23 

.04060 

.04232 

.04507 

.04786  i  .05103 

.05378 

.05668 

.06009 

23 

24 

.04069 

.04241 

.04576 

.04795   .05112 

.05388 

.05678 

.06020 

24 

25 

.04077 

.04250 

.04585 

.04805  :  .05122 

.05398 

.05687 

.060:30 

25 

26 

.04085 

.04259  i 

.04593 

.04815  i  .05131 

.05408 

.05697  !  .06041 

26 

27 

.04093 

.04268 

.04602   .04824  1  .05140   .05418 

.05707   .06052 

27 

28 

.04102 

.04277 

.04611 

.04834  i  .05149  i  .05429 

.05716 

.06063 

28 

29 

.04110 

.04286 

.04620 

.04843  1  .05158 

.05439 

.05726 

.06074 

29 

30 

.04118 

.04295 

.04628 

.04853  ; 

.05168 

.05449 

.05736 

.06085 

30 

31 

.04126 

.04304 

.04637 

.04863  ' 

.05177 

.05460 

.05746 

.06096 

31 

32 

.04135 

.04313 

.04646 

.04872 

.05186 

.05470 

.05755   .06107 

32 

33 

.04143 

.04322 

.04655 

.04882 

.05195 

.05480 

.05765 

.06118 

33 

34 

.04151 

.04:331 

.04663 

.04891 

.05205 

.05490 

.05775 

.06129 

34 

35 

.04159 

.04340 

.04672 

.04901 

.05214 

.05501 

.05785 

.06140 

35 

36 

.04168 

.04349 

.04681 

.04911 

.05223 

.05511 

.05794 

.06151 

36 

37 

.04176 

.04358 

.04690 

.04920 

.05232 

.05521 

.05804 

.06162 

37 

38 

.04184 

.04367 

.04699 

.04930 

.05242 

.05532 

.05814 

.06173 

38 

30 

.04193 

.04376 

.04707 

.04940  i 

.05251 

.C5542 

.05824 

.06184 

39 

40 

.04201 

.04385  ; 

.04716 

.04950 

.05260 

.05552 

.05833 

.06195 

40 

41 

.04209 

.04394 

.04725 

.04959 

.05270 

.05563 

.05843 

.06206 

41 

42 

.04218 

.04403 

.04734 

.04969 

.05279 

.05573 

.05853 

.06217 

42 

43 

.04226 

.04413 

.04743 

.04979  ! 

.05288 

.05584 

.05863 

.06228 

43 

44 

.04234 

.04422  i 

.04752 

.04989  i 

.05298 

.05594 

.05873 

.06239 

44 

45 

.04243 

.04431 

.04760 

.04998 

.05307 

.05604 

.05882 

.06250 

45 

46 

.04251 

.04440  ; 

.04769 

.05008 

.05316 

,.05615 

.05892 

.06261 

46 

47 

.04260 

.04449 

.04778 

.05018  | 

.05326 

.05625 

.05902 

.06272 

47 

48 

.04268 

.04458 

.04787 

.05028  ! 

.05335 

.05636 

.05912 

.06283 

48 

49 

.04276 

.04468 

.04796 

.05088 

.05344 

.05646 

.05922 

.00295 

49 

50 

.04285 

.04477 

.04805 

.05047 

.05354 

.05657 

.05932 

.06306 

50 

51 

.04293 

.04486 

.04814 

.05057 

.05363 

.05667 

.05942 

.06317 

51 

52 

.04302 

.04495  ' 

-  .04823 

.0.5067 

.05373 

.0567S 

.05951 

.06328 

52 

53 

.04310 

.04504 

.04832 

.06077 

.03383 

.05688 

.05961 

.06339 

53 

54 

.04319 

.04514 

.04841 

.05087  ll  .05391 

.05699  1 

.05971 

.06350 

54 

55 

.04327 

.04523 

.04850 

.05097   .05401 

.05709  1  .05981 

.06362 

55 

56 

.04,336 

.04532 

.04858 

.05107   .05410 

.05720  1  .05991 

.06373 

56 

57 

.04344 

.04541 

.04867 

.05116 

.05420 

.05730   .06001 

.06384 

57 

58 

.04353 

.04551  i 

.0-1876 

.05126 

.05429 

.05741 

.06011 

.06895 

58 

59 

.  04361 

.045(50 

.04885 

.05136 

.05439 

.05751 

.06021 

.0640? 

59 

60 

.04370 

.04569 

.04894 

.05146  ||  .05448 

.05762   .06031 

.06418  i  60 

325 


TABLE  XIII.— VERSINES  AND  EXSECANTS. 


2( 

)° 

2] 

P 

* 

5° 

2( 

J° 

/ 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

1 

0 

.0(5031 

.06418 

.06642 

.07115 

.07282 

.07853 

.07950 

.06686 

0 

1 

.06011 

.06429  I 

.06652 

.07126 

.07293 

.07866 

.07961 

.OS649 

1 

2 

.06051 

.06440 

.06663 

.07138 

.07303 

.07879 

.07972 

.08603 

2 

3 

.06061 

.06452 

.06673 

.07150 

.07314 

.07892 

.07984 

.08676 

3 

4 

.06071 

.06463 

.06684 

.07102 

.07325 

.07904 

.07995 

.08690 

4 

5 

.06081 

.06474 

.06694 

.07174 

.07336 

.07917 

.08006 

.08703 

5 

6 

.06091 

.06486 

.06705 

.07186 

.07347 

.07930 

.08018 

.08717 

6 

7 

.06101 

.06497 

.06715 

.07199 

.07358 

.07943 

.08029 

.08730 

7 

8 

.06111 

.06508 

.06726 

.07211 

I  .07369 

.07955 

.08041 

.08744 

8 

9 

.06121 

.06520 

.06736 

.07*23 

.07380 

.07968 

.08052 

.08757 

9 

10 

.06131 

.06531 

.06747 

.07235 

i  .07391 

.07981 

.08004 

.08771 

10 

11 

.06141 

.06.542 

.06757 

.07247 

.07402 

.07994 

.08075 

.08784 

11 

12 

.06151 

.06554 

.06768 

.07259 

.07413 

.08006 

.08086 

.08798 

12 

13 

.06161 

.06565 

.06778 

.07271 

.07424 

.08019 

.08098 

.08811 

13 

14 

.06171 

.06577 

.06789 

.07283 

.07435 

.08032 

.08109 

.08825 

14 

15 

.06181 

.06588 

.06799 

.07295 

.07446 

.08045 

.08121 

.08839 

15 

16 

.06191 

.06600 

.06810 

.07307 

!  .07457 

.08058 

.08132 

.08852 

1(5 

IT 

.06201 

.06611 

.06820 

.07320 

.07468 

.08071 

.08144 

.08806 

17 

18 

.06211 

.06622 

.06831 

.07332 

.0747'9 

.08084 

.08155 

.08880 

18 

19 

.06221 

.06634 

.06841 

.07344 

.07490 

.08097 

.08167 

.08893 

19 

20 

.06231 

.06645 

.06852 

.07356 

.07501 

.08109 

:  .08178 

.08907 

20 

21 

.06241 

.06657 

.06863 

.07368 

.07512 

.08122 

.08190 

.08921 

21 

22 

.06252 

.06668 

.06873 

.07380 

.07523 

.08135 

.08201 

.08934 

22 

23 

.06262 

.06680 

.06884 

.07393 

.07534 

.08148 

.08213 

.08948 

>  23 

24 

.06272 

.06691 

.06894 

.07405 

.07545 

.08161 

.08225 

.08962 

24 

25 

.06282 

.06703 

.06905 

.07417 

.07556 

.08174 

.08236 

.08975 

25 

26 

.06292 

.06715 

.06916 

.07429 

.07568 

.08187 

.08248 

.08989 

26 

27 

.06302 

.06726 

.06926 

.07442 

.07579 

.08200 

.08259 

.09003 

27 

28 

.06312 

.06738 

.06937 

.07454 

.07590 

.08213 

.08271 

.09017 

23 

29 

.06323 

.06749 

!  .06948 

.07466 

.07601 

.08226 

.0828-2 

.09030 

29 

30 

.06333 

.06761 

.06958 

.07479 

.07612 

.08239 

.08294 

.09044 

30 

31 

.06343 

.06773 

.06969 

.07491 

.07623 

.08252 

.08306 

.09058 

31 

32 

.06353 

.06784 

.06980 

.07503 

.07634 

.08265 

.08317 

.01)072 

32 

33 

.06363 

.06796 

.06990 

.07516 

.07645 

.08278 

.08329 

.09080 

33 

34 

.06374 

.06807 

.07001 

.07528 

.07657 

.08291 

.08340 

.09099 

34 

35 

.06384 

.06819 

i  .07012 

.07540 

.07668 

.08305 

.08352 

.09113 

35 

36 

.06394 

.06831 

i  .07022 

.07553 

.07'079 

.08318 

.08364 

.09127 

36 

37 

.06404 

.06843  , 

!  .07033 

.07565 

;  .07090 

.08331 

.08375 

.09141 

37 

38 

.06415 

.06854 

.07044 

.07578 

.07701 

.08344 

.08387 

.09155 

38 

39 

.06425 

.06866 

.07055 

.07590 

i  .07713 

.08357 

.08399 

.09169 

39 

40 

.06435 

.06878 

.07065 

.07602 

.07724 

.08370 

.08410 

.09183 

40 

41 

.06445 

.06889 

.07076 

.07615 

.07735 

.08383 

.08422 

.09197 

41 

42 

.06456 

.06901 

.07087 

.07627 

.07746 

.08397 

.08434 

.09211 

42 

43 

.06466 

.06913 

.07098 

.07640 

.07757 

.08410 

.08445 

.09224 

43 

44 

.06476 

.06925 

.07108 

.07652 

.07769 

.08423 

.08457 

.09238 

44 

45 

.06486 

.06936  I 

.07119 

.07665 

1  .07780 

.08436 

.08469 

.09252 

45 

46 

.06497 

.06948 

.07130 

.07677 

.07791 

.08449  ! 

.08481 

.09266 

4(5 

47 

.06507 

.06960 

.07141 

.07690 

.07802 

.08463 

.08492 

.09280 

47 

48 

.06517 

.06972 

.07151 

.07702 

.07814 

.08476 

.08.504 

.09294 

4S 

49 

.06528 

.06984 

.07162 

.07715 

.07835 

.08489 

.08516 

.09308 

49 

50 

.06538 

.06995 

.07173 

.07727 

.07836 

.08503 

.08528 

.09323 

60 

51 

.06548 

.07007 

.07184 

.07740 

.07848 

.08516 

.08539 

.09&37 

51 

52 

.06559 

.07019 

.07195 

.07752 

.07859 

.08529 

.08551 

.09351 

52 

53 

.06569 

.07031  1 

.07206 

.07765 

.  .07870 

.08548 

.08563 

.09305 

53 

54 

.06580 

.07043  ; 

.07216 

.07778 

.07881 

.08556 

.08575 

.09379 

54 

55 

.06590 

.07055  : 

.07227 

.07790 

.  07893 

.0856!)  : 

.08586 

.09393 

55 

56 

.06600 

.07067 

.07238 

.07803 

.07904 

.08582  1 

.08598 

.09407 

50 

57 

.06611 

.07079 

.07249 

.07811) 

.07915 

.08596 

.08610 

.09421 

57 

58 

.06621 

.0701)1 

.07260 

.07828 

!  .07927 

.08(509 

.08622 

.09435 

58 

59 

.06632 

.07103 

.07271 

.07841 

.079:38 

.0862;? 

.08634 

.09449 

59 

60 

.06642 

.07115 

.072«2 

.07853 

.07950 

.08036  ! 

.06645 

.09404 

60 

326 


TABLE  XIII.-VERSINES  AND  EXSECANTS. 


/ 

24° 

25°         26° 

27° 

/ 

Vers. 

Exsec. 

Vers. 

Exsec. 

Yers. 

Exsec. 

Vers. 

Exsec. 

0   .08645 

.094(54 

.09369 

.10338  | 

.10121 

.11260  ! 

.10899 

.12233 

0 

1   .08057 

.09478 

.09382 

.10353  1 

.10133 

.11276 

.10913 

.1.2249 

1 

2 

.08069 

.09492 

.09394 

.10368 

.10146 

.11292 

.10926 

.12266 

2 

0 

.08681 

.09506 

.09106 

.10383  i 

.10159 

.11308 

.10939 

.12283 

3 

4 

.08693 

.09520 

.09418 

.10398 

.10172 

.11323  ! 

.10952 

.12299 

4 

5 

.08705   .09535 

.09431 

.10413 

.10184 

.11389 

.10965 

.12316 

5 

0 

.08717 

.09549 

.09443 

.10428  i 

.10197 

.11355 

.10979 

.12333 

G 

.08728 

.09563 

.09455 

.10443  j 

.10210 

.11371 

.10992 

.12349 

7 

8 

.08740 

.09577 

.09468 

.10458 

.10223 

.11387 

.11005 

.12366 

8 

9 

.08752 

.09592 

.09480 

.10473 

.10236 

.11403 

.11019 

.12383 

9 

10 

.087G4 

.09606 

.09493 

.10488 

.10248 

.11419 

i  .11032 

.12400 

10 

11 

.08776 

.09620 

.09505 

.10503 

.10261 

.11435 

.11045 

.12-116 

11 

12 

.08788 

.09635 

.09517 

.10518 

.10274 

.11451 

.11058 

.12433 

12 

13 

.08800 

.09649 

.09530 

.10533 

.10287 

.11467 

.11072 

.12450 

13 

14 

.08812 

.09663 

.09542 

.10549 

.10300 

.11483 

.11085 

.12467 

14 

15 

.08824 

.09678 

.09554 

.10564 

.10313 

.11499 

.11098 

.12484 

15 

1(5 

.08836 

.09692 

.09567 

.10579 

.10326 

.11515 

.11112 

.12501 

16 

17 

.08848  !  .09707 

.09579 

.10594 

.10338 

.11531 

.11125 

.12518 

17 

18 

.08860  i  .09721 

.09592 

.10609 

.10351 

.11547 

.11138 

.12534 

18 

19 

.08872 

.09735 

.09604 

.10625 

.10364 

.11563 

.11152 

.12551 

19 

20 

.08884 

.09750 

.09617 

.10640 

.10377 

.11579 

.11105 

.12568 

20 

21 

.08896 

.09764 

.09629 

.10655 

.10390 

.11595 

.11178 

.12585 

21 

.089G3 

.0977!) 

.09042 

.10670 

.10403 

.11611 

.11198 

.12602 

22 

23 

.08920 

.097'93 

.09654 

.10686 

.10416 

.11627 

.11205 

.12619 

23 

24 

.08932 

.09808 

.09666 

.10701 

.10429 

.11643 

.11218 

.12636 

24 

25 

.08944 

.09822 

.09679 

.10716 

.10442 

.11659 

.11232 

.12653 

25 

26 

.08956 

.09837 

.09691 

.10731 

.10455 

.11675 

.11245 

.12670 

26 

27 

.08968 

.09851 

.097'04 

.10747 

.10468 

.11691 

.11259 

.12687 

27 

2S 

.08980 

.09866 

.09716 

.10762 

.10481 

.11708 

.11273 

.12704 

28 

29 

.08992 

.09880 

.09729 

.10777 

.10494 

.11724 

.11285 

.12721 

29 

30 

.09004 

.09895 

.09741 

.10793 

.10507 

.11740 

.11299 

.12738 

30 

31 

.09016 

.09909 

.09754 

.10808 

.10520 

.11756 

.11312 

.12755 

31 

32 

.09028 

.09924 

.09767 

.10824 

.10533 

.11772 

.11326 

.12772 

32 

33 

.09040 

.09939 

.09779 

.10839 

.10546 

.11789 

.11339 

.12789 

33 

34 

.09052 

.09953 

.09792 

.10854 

.10559 

.11805 

.11353 

.12807 

34 

35 

.09064 

.09968 

.09804 

.10870 

.1057-2 

.11821 

.11366 

.12824 

35 

36 

.09076 

.09982 

.09817 

.10885 

.10585 

.11838 

.11380 

.12841 

36 

37 

.09089 

.09997 

.09829 

.10901 

.10598 

.11854 

.11393 

.12858 

37 

38 

.09101 

.10012 

.09842 

.10916 

.10611 

.11870 

.11407 

.12875 

38 

39 

.09113 

.10026 

.09854 

.10932 

.10624 

.11886 

.11420 

.12892 

39 

40 

.09125 

.10041 

.09867 

.10947 

.10(537 

.11903 

.11434 

.12910 

40 

41 

.09137 

.10055 

.09880 

.10963 

.10650 

.11919 

.11447 

.12927 

41 

42 

.09149 

.10071 

.09892 

.10978 

.10663 

.11936 

.11461 

.12944 

42 

4:} 

.09161 

.10085 

.09905 

.10994 

.10676 

.11952 

.11474 

.12961 

43 

44 

.09174 

.10100 

.09918 

.11009 

.10689 

.11968 

.11488 

.12979 

44 

45 

.09186 

.10115 

.09930 

.11025 

.10702 

.11985 

.11501 

.12996 

4") 

46 

.09198 

.10130 

.09943 

.11041 

.10715 

-12001 

.11515 

.13013 

46 

47 

.09210 

.10144 

.09955 

.11056 

.10728 

.12018 

.11523 

.13031 

47 

48 

.09222 

.10159 

.09968 

.11072 

.10741 

.12034 

.11542 

.13048 

48 

49 

.09234 

.10174 

.09981 

.11087 

.10755 

.12051 

.11555 

.13065 

49 

50 

.09247 

.10189 

.09993 

.11103 

.10768 

.12067 

.11569 

.13083 

50 

51 

.09259 

.10204 

.10006 

.11119 

.10781 

.12084 

.11583 

.13100 

51 

52 

.09271 

.10218 

-.10019 

.11134 

.10794 

.12100 

.11596 

.13117 

52 

53 

.09283 

.10233 

.10032 

.11150 

.10807 

.12117 

.11610 

.13135 

53 

54 

.09296 

.10248 

.10044 

.11166 

.10820 

.12133 

.11623 

.13152 

54 

55 

.09308 

.10263 

.10057 

.11181 

.10833 

.12150 

.11637 

.13170 

55 

56 

.09320 

.10278 

.10070 

.11197 

.10847 

.12166 

.11651 

.13187 

56 

57 

.09332 

.10293 

.10082 

.11213 

.10860 

.12183 

.11664 

.13205 

57 

58 

.09345 

.10308 

.10095 

.11229 

.10873 

.12199 

.11678 

.13222 

58 

59 

.09357 

.10323 

.10108 

.11244 

.10886 

.12216 

!  .11692 

.13240 

59 

60 

.09369 

.10338 

.10121 

.11260 

.10899 

.12233 

i  .11705 

.13257 

60 

TABLE  XIII.-VERSINES  AND  EXSECANTS. 


' 

28° 

29° 

30° 

31° 

/ 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers.  |  Exsec. 

Vers. 

Exsec. 

0 

.11705 

.13257 

.12538 

.14335 

.13397 

.15470 

.14283 

.16603 

~T 

1 

.11719 

.13275 

.12552 

.14354 

.13412 

.15489 

.14298 

.10084 

i 

2 

.11733 

.13292 

.12566 

.14372 

.13427 

.15509 

.14313 

.16704 

2 

ft 

.11746 

.13310 

.12580 

.14391 

.13441 

.15528 

.14328 

.16725 

3 

4 

,11760 

.13327 

.12595 

.14409 

.13456 

.15548 

.14343 

.16745 

4 

5 

.11774 

.13345 

.12609 

.14428 

.13470 

.15567 

.14358 

.16766 

5 

6 

.11787 

.13362 

.12623 

.14446 

.13485 

.15587 

.14373 

.16786 

6 

7 

.11801 

.13380 

.12637 

.14465 

.13499 

.15606 

.14388 

.16806 

7 

8 

.11815 

.13398 

.12651 

.14483 

.13514 

.15026 

.14)03 

.16827 

8 

9 

.11828 

.13415 

.12665 

.14502 

.13529 

.15645 

.14418 

.10848 

9 

10 

.11842 

.13433 

.12679 

.14521 

.13543 

.15665 

.14488 

.16868 

10 

11 

.11856 

.13451 

.12694 

.14539 

.13558 

.15684 

.14119 

.16889 

11 

12 

.1187'0 

.13468 

.12708 

.14558 

.13573 

.15704 

.14464 

.10909 

12 

13 

.11883 

.13486 

.12722 

.14570 

.13587 

.15724 

.14479 

.16930 

13 

14 

.11897 

.1&504  1 

.12736 

.14595 

.13602 

.15743 

.14194 

.10950 

14 

15 

.11911 

.13521  1 

.12750 

.14614 

.13016 

.15763 

.14509 

.16971 

15 

16 

.11925 

.13539 

.12705 

.14632 

.13631 

.15782 

.115:24 

.10092 

16 

17 

.11938 

.13557 

.12779 

.14651 

.13646 

.15802 

.14539 

.17012 

17 

18 

.11952 

.13575 

.12793 

.14070 

.13660 

.15823 

.14554 

.17033 

18 

19 

.11966 

.13593 

.12807 

.14GS9 

.13675 

.15841 

.11509 

.17  .11 

19 

20 

.11980 

.13610 

.12822 

.14707 

.13690 

.15861 

.14584 

.17075 

20 

21 

.11994 

.13628 

.12836 

.14726 

.13705 

.15881 

.14599 

.17095 

21 

22 

.12007 

.13646 

.12850 

.14745 

.13719 

.15901 

.14615 

.17116 

22 

23 

.12021 

.13664 

.12864 

.14764 

.13734 

.15920 

.14630 

.17137 

23 

24 

.12035 

.13682 

.12879 

.14782 

.13749 

.15940 

.14645 

.17158 

24 

25 

.12049 

.13700 

.12893 

.14801 

.13763 

.15960 

.14600 

.17178 

25 

26 

.12063 

.13718 

.12907 

.14820 

.13778 

.15980 

.14675 

.17199 

26 

27 

.12077 

.13735 

.12921 

.14839 

.13793 

.16000 

.14690 

.17320 

27 

28 

.12091 

.13753 

.12936 

.14858 

.136C8 

.10019 

.14706 

.17.041 

28 

29 

.12104 

.13771 

.12950 

.14877 

.13822 

.16039 

.14721 

.17262 

29 

30 

.12118 

.13789 

.12964 

.14896 

.13837 

.16059 

.14736 

.17283 

30 

31 

.12132 

.13807 

.12979 

.14914 

.13852 

.16079 

.14751 

.17304 

31 

32 

.12146 

.13825 

.12993 

.14933 

.13867 

.16099 

.14766 

.17325 

32 

33 

.12160 

.13843 

.13007 

.1495? 

.13881 

.16119 

.14782 

.17346 

33 

34 

.12174 

.13861 

.13022 

.14971 

.13896 

.16139 

.14797 

.17307 

34 

35 

.12188 

.13879 

.13036 

.14990 

.13911 

.16159 

.1481:3 

.17388 

35 

36 

.12202 

.13897 

.13051 

.15009 

.13926 

.16179 

.14827 

.17409 

36 

37 

.12216 

.13916 

.13065 

.15028 

.13941 

.16199 

.14843 

.17-430 

37 

38 

.12230 

.13934 

.13079 

.15047 

.13955 

.16219 

.14858 

.17451 

38 

39 

.12244 

.13952 

.13094 

.15066 

.13970 

.16239 

.14873 

.17472 

39 

40 

.12257 

.13970 

.13108 

.15085 

.13985 

.16259 

.14888 

.17493 

40 

41 

.12271 

.13988 

.13122 

.15105 

.14000 

.16279 

.14004 

.17514 

41 

42 

.12285 

.14006 

.13137 

.15124 

.14015 

.1  0:21)9 

.14919 

.17535 

42 

43 

.12299 

.14024 

.13151 

.15143 

.14030 

.16319 

.14934 

.17556 

43 

44 

.12313 

.14042 

.13166 

.15102 

.14044 

.16339 

.14949 

.17577 

44 

45 

.12327 

.14061 

.13180 

.15181 

.14059 

.16359 

.14905 

.17598 

45 

46 

.12341 

.14079 

.13195 

.15200 

.14074 

.16380 

.14980 

.17620 

46 

47 

.12355 

.14097 

.13209 

.15219 

.11089 

.16400 

.14995 

.17641 

47 

48 

.12369 

.14115 

.13228 

.15239 

.14104 

.16420 

.15011 

.17662 

48 

49 

.12383 

.14134 

.13238 

.15258 

.14119 

.16440 

.150-26 

.17688 

49 

50 

.12397 

.14152 

.13252 

.15277 

.141:54 

.10460 

.15041 

.17704 

50 

51 

.12411 

.14170 

.13267 

.15296 

.14149 

.16481 

.15057 

.17726 

51 

52 

.12425 

.14188 

.18281 

.15315 

.14164 

.16501 

.15072 

.17747 

52 

53 

.12439 

.14207 

.13296 

.15335 

.14179 

.16521 

.15087 

.17768 

53 

54 

.12454 

.14225 

.13310 

.15354 

.14194 

.16541 

.15103 

.17790 

54 

55 

.12468 

.14243 

.18326 

.15373 

.14208 

.16562 

.15118 

.17811 

55 

56 

.12482 

.14262 

.13339 

.15393 

.14223 

.16582 

.15184 

.17833 

56 

-  57 

.12496 

.14280 

.13864 

.15412 

.14888 

.16602 

.15149 

.17854 

57 

58 

.12510 

.14299 

.18868 

.16431 

.14253 

.100:23 

.15164 

.17875 

58 

59 

.12524 

.14317 

.13383 

.15451 

.14268 

.16643 

.15180 

.17896 

59 

60 

.12538 

.14335 

.13397 

.15470 

.14283 

.16663 

.15195 

.17918 

60 

TABLE  XIII.— VERSINES  AND  EXSECANTS. 


32a 

33° 

34° 

35° 

/ 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.15195 

.17918 

.16133 

.19236 

.17096 

.20622 

.18085 

.22077 

0 

1 

.15211 

.17939 

.16149 

.192.59 

.17113 

.20645 

.18101 

.22102 

1 

2 

.15226 

.17961 

.16165 

.19281 

.17129 

.20669 

.18118 

.22127 

2 

3 

.15241 

.17982 

.16181 

.19304 

.17145 

.20693 

.18135 

.22152 

3 

4 

.15257 

.18004 

.16196 

.19327 

.17161 

.20717 

.18152 

.22177 

4 

5 

.15272 

.18025 

.16212 

.19349 

.17178 

.20740 

.18168 

.22202 

5 

c 

.  15288 

.18047 

.16228 

.19372 

.17194 

.20764 

.18185 

.22227 

6 

7 

.15303 

.18068 

.16244 

.19394 

.17210 

.20788 

.18202 

.22252 

7 

8 

.15319 

.18090 

.16260 

.19417 

.17227 

.20812 

.18218 

.22277 

8 

9 

!  15334 

.18111 

.16276 

.19440 

.17243 

.20836 

.18235 

.22302 

9 

10 

.15350 

.18133 

.16292 

.19463 

.17259 

.20859 

.18252 

.22327 

10 

11 

.15885 

.18155 

.16308 

.19485 

.17276 

.20883 

.18269 

.22352 

11 

12 

.15381 

.18176 

.16324 

.19508 

.17292 

.20907 

.18286 

.22377 

12 

13 

.15396   .18198 

.16340   .19531 

.17308 

.20931 

.18302 

.22402 

13 

14 

.15412  !  .18220 

.16355   .19554 

.17325   .20955 

.183-9 

.22428 

14 

15 

.15427   .18241 

.16371 

.1957'6 

.17341 

.20979 

.18336 

.22453 

15 

16 

.15443  i  .18263 

.16387 

.19599 

.17357 

.21003 

.18353 

.22478 

16 

17 

.15458   .18285 

.16403 

.19622 

.17374 

.21027 

.18369 

.22503 

17 

18 

.15474 

.18307 

.16419 

.19645 

.17390 

.21051 

.18386 

.22528 

18 

19 

.15489 

.18328 

.16435 

.19668 

.17407 

.21075 

.18403 

.22554 

19 

20 

.15505 

.18350 

.16451 

.19691 

.17423 

.21099 

.18420 

.22579 

20 

21 

.15520   .18372 

.16467 

.19713 

.17439 

.21123 

.18437 

.22604 

21 

22 

.15536 

.18394 

.16483 

.19736   .17456 

.21147 

.18454 

.22629 

22 

23 

.15552 

.18416 

.16499 

.19759  i  .17472 

.21171 

.18470   .22655 

23 

24 

.15567 

.18437 

.16515 

.19782   .17489 

.21195 

.18487   .22680 

24 

25 

.15583 

.18459 

.16531 

.19805   .17505 

.21220 

.18504   .227'06 

25 

26 

.15598   .18481 

.16547 

.19828   .17522 

.21244 

.18521   .22731 

26 

27 

.15614  ;  .18503 

.16563 

.19851 

.17538 

.21268 

.18538   .227'56 

27 

28 

.15630 

.18525 

.16579 

.19874 

.17554 

.21292 

!  .18555  :  .22782 

28 

29 

.15645 

.18547 

.16595 

.19897   .17571 

.21316 

I  .18572   .22807 

29 

30 

.15661 

.18569 

.16611 

.19920 

.17587 

..21341 

.18588   .22833 

30 

31 

.15676 

.18591 

.16627 

.19944  ! 

.17604 

.21365 

.18605   .22858 

31 

32 

.15692 

.18613 

.16644   .19967  i 

.17620 

.21389 

.18622   .22884 

32 

33 

.15708 

.18635 

.16660   .19990 

.17037   .21414 

.18639   .22909 

33 

34 

.15723  i  .18657 

.16676   .20013 

.17653   .21438 

.18656   .22935 

34 

35 

.15739  !  .18679 

.16692   .20036 

.17670   .21462 

.18673 

.22960 

35 

36 

.15755   .18701 

.16708   .20059 

.17686 

.21487 

.18690 

.22986 

36 

37 

.15770   .18723 

.16724   .20083 

.17703 

.21511 

.18707 

.23012 

37 

38 

.15786   .18745 

.16740   .20106 

.17719 

.21535 

.18724 

.23037 

38 

39 

.15802  I  .18767 

.16756   .20129 

.17738 

.21560 

.18741 

.23063 

39 

40 

.15818   .18790 

.16773   .20152 

.17752 

.21584 

.18758 

.23089 

40 

41 

.15833   .18812 

.16788  .20176 

.17769 

.21609 

.18775 

.23114 

41 

42 

.15849  i  .18834 

.16805   .20199   .17786 

.21633 

.18792 

.23140 

42 

43 

.15865 

.18856 

.16821   .2<  )•-•_>   .17802 

.21058 

.18809 

.23166 

43 

44 

.15880 

.18878 

.16837   .2021(3 

.17819 

.21682 

.18826 

.23192 

44 

45 

.15896 

.18901 

.16853   .20269 

.17835 

.21707 

.18843 

.23217 

45 

46 

.1.7.)  12   .18923 

.16869   .20292 

.17852- 

.21731 

.18860 

.23243 

46 

47 

.159-3-}   .18945 

.16885   .2C316  i 

.17868 

.21756 

.18877 

.23269 

47 

48 

.15943  !  .18967 

.16902   .20:339 

.17885 

.21781 

.18894 

.23295 

48 

49 

.15950   .18990 

.16918   .20363 

.17902 

.21805 

.18911 

.233.21 

49 

50 

.15975   .19012 

.16934  -.20386 

.17918 

.21830 

48928 

.23347 

50 

51 

.15991 

.19034 

.16950 

.20410 

.17935 

.21855 

.18945 

.23373 

51 

52 

.16006   .19057 

.16966 

.20433 

.17952 

.21879 

.18962 

.23399 

52 

53 

.16022   .19079 

.16983 

.20157   .17968 

.21904 

.18979 

.23424 

53 

54 

.160:38   .19102  i 

.16999 

.20480  j  .17985 

.21929 

.18996 

.23450 

54 

55 

.16054 

.19124 

.17015 

.20501  :  .18001 

.21953 

.19013 

.23476 

55 

56 

.16070 

.19146 

.17031 

.20527  1  .18018 

.21978 

.19030 

.23502 

56 

57 

.16085 

.19169 

.17047 

.20551 

.18035 

.22003 

.19047 

.23529 

57 

58 

.16101 

.19191 

.17064 

.30575 

.18051 

.22028 

.19064 

.23555 

58 

59 

.16117   .19214 

.  17080 

.20598  il  .18068 

.22053 

.19081 

.23581 

59 

60 

.16133  1  .19236 

.17096 

.20622  1  .18085  1  .22077 

.19098 

.23607 

60 

TABLE  XIII.— VEKSINES  AND  EXSECANTS. 


/ 

36° 

37° 

38° 

39° 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers.  Exsec. 

~ 

.19098 

.23007 

.201&6 

.25214 

.21199 

.26902 

.22285 

.28676 

0 

1 

.19115 

.23633 

.20154 

.25241 

.21217 

.26931 

.32304 

.287'06 

1 

2 

.19133 

.23659 

.20171 

.25269 

.21235 

.26960 

.22322 

.28737 

2 

3 

.19150 

.23685 

.20189 

.25296 

.21253 

.26988 

.22340 

.28767 

3 

4 

.19167 

.23711 

.20207 

.25324 

.21271 

.27017 

.22359 

.28797 

4 

5 

.1918! 

.23738 

.20224 

.25351 

.21289 

.27046 

.22377 

.28828 

5 

6 

.19201 

.23764 

.20242 

.25379 

.21307 

.27075 

.22395 

.28853 

6 

7 

.19218 

.23790 

.20259 

.25406 

.21324 

.27104 

.22414 

.28889 

7 

8 

.19235 

.23816 

.20277 

.25434 

.21342 

.27133 

.22432 

.28919 

8 

9 

.19252 

.23843 

.20294 

.25462 

.213(50 

.27162 

.22450 

.28950 

9 

10 

.19270 

.23869 

.20312 

.25489 

.21378 

.27191 

.22409 

.28980 

10 

n 

.19287 

.23895 

.20329 

.25517 

.21396 

.27221 

.22487 

.29011 

11 

13 

.19304 

.23922 

.20347 

.25545 

.21414 

.27250 

.22506 

.29042   12 

13 

.19321 

.23948 

.20365 

.25572 

.21432 

.27'27'D 

.22521 

.29072   13 

14 

.19338 

.23975 

.20382 

.25600 

.21450 

.27308 

.22543 

.29103   14 

15 

.19356 

.24001 

.20400 

.25628 

.21468 

.27'337 

.225(51 

.29133   15 

16 

.19373 

.24028  !  .20417 

.25656 

.21486 

.373(56 

.2257'!) 

.29164   16 

17 

.19390 

.24054   .20135 

.25683 

.21504 

.37396 

.32598 

.29195   17 

18 

.19407 

.24081 

.20453 

.25711 

.21522 

.27425 

.22616 

.29226 

18 

•  19 

.19424 

.24107 

.20470 

.25739 

.21540 

.27454 

.22634 

.29256 

19 

20 

.19442 

.24134  II  .20488 

.25767 

.21558 

.27-483 

.22653 

.29287 

20 

21 

.19459 

.24160   .20506 

.25705 

.21576 

.27513 

.22671 

.29318 

21 

22 

.19476 

.24187  !  .20523 

.35823 

.21595 

.37543 

.22690 

.29349 

22 

23 

.19493 

.24213 

.20541 

.25851 

.21613 

.2;5;2 

.22708 

.29380 

23 

24 

.19511 

.24240 

.20559 

.25879  ; 

.21631 

.27(501  , 

.227'27 

.29411 

24 

25 

.19528 

.24267 

.20576 

.25907  , 

.21649 

.27630  1 

.22745 

.29442 

25 

26 

.19545 

.24293 

.20594 

.25935  t 

.21667 

.27660 

.22764 

.29473  i  26 

27 

.195(52 

.24320 

.20612 

.25963 

.21685 

.27689 

.22782 

.29504 

27 

28 

.19580 

.24347 

.  20629  , 

.25991 

.21703 

.27719 

.22801 

.29535 

38 

29 

.19597 

.24373 

.20647 

.26019 

.21721 

.27748 

.22819 

.29568 

2:) 

30 

.19614. 

.24400 

.20665 

.26047 

.21739 

.27778  i 

.22838 

.28597 

SO 

81 

.19632 

.24427 

.20682 

.26075 

.21757 

.27807 

.22856 

.29628 

31 

33 

.19649 

.24454 

.20700 

.26104  : 

.217:5 

.27837 

.228:5 

.29659 

33 

33 

.19666 

.24481 

.20718 

.26132  | 

.21794 

.27867 

.22893 

.29(590 

83 

34 

.1J684 

.24508 

.20736 

.26160 

.21812 

.27896  j 

.22912 

.297-21 

31 

35 

.19701 

.24534 

.20753 

.26188  i 

.21830 

.27926  I 

.22030 

.39752 

35 

36 

.19718 

.24561 

.20771 

.26216 

.21848 

.27956 

.22949 

.29781 

36 

37 

.19736 

.24588 

.20789 

.26245 

.21866 

.37985  ! 

.22967 

.29815 

£7 

38 

.19753 

.24615 

.20807 

.26273  I 

.21884 

.28015 

.22986 

.29846 

33 

39 

.19770 

.21642 

.20824 

.26301  , 

.21902 

.28045 

.23004 

.39877 

39 

40 

.19788 

.24669 

.20842 

.26330 

.21921 

.28075 

.23023 

.29909 

40 

41 

.19805 

.24696 

.20860 

.26358 

.21939 

.28105 

.23041 

.29910 

41 

.19822 

.34733 

.2087'8 

.26387 

.211)57 

.28134 

.23060 

.39971 

42 

43 

.19840 

.24750 

.20895 

.26415 

.21975 

.28164 

.2307'9 

.30003 

43 

44 

.19857 

.24777 

.20913 

.26443 

.21993 

.28194 

.23097 

.30034 

44 

45 

.19875 

.24804 

.20931 

.26472 

.22012 

.28224 

.23116 

.30066 

45 

46 

.19892 

.24832 

.20949 

.26500 

.22030 

.28254  i 

.231:34 

.30097 

46 

47 

.19909 

.34859 

.20967 

.26529 

.22048 

.28284  ] 

.23153 

.30129 

47 

48 

.19927 

.24886 

.20985 

.26557 

.22066 

.28314  1 

.23172 

.301(50 

43 

49 

.19944 

.21913 

.21002 

.26586 

.22084 

.28344 

.23190 

.30192 

49 

50 

.19962 

[34940 

.21020 

.26615  i 

.22103 

.28374 

.23209 

.30223 

50 

51 

.1997-9 

.24967 

.21038 

.26643 

.22121 

.28404  i 

.23228 

.30255 

51 

52 

.19997 

.24995 

.21056 

.26G7'2 

.22139 

.28434 

.23246 

.30287 

52 

53 

.20014 

.25022 

.21074 

.26701 

.22157 

.28464 

.23265 

.30318 

53 

54 

.20032 

.25049  ! 

.21092 

.26729 

.22176 

.28495 

.23283 

.30350 

54 

55 

.20049 

.25077  ! 

.21109 

.26758 

.22194 

.28525 

.23302 

.30382 

55 

56 

.20066 

.25104  ; 

.21127 

.2(5787 

.33313 

.28555  ! 

.23321 

.30413 

56 

57 

.20084 

.25131 

.21145 

.26815 

.22231 

.28585 

.23339 

.30445 

57 

58 

.20101 

.25159 

.21163 

.26844 

22219 

.38(51.-) 

133358 

.30477 

58 

59 

.20119 

.25186  | 

.21181 

.26873 

!  33267 

.28646 

.23377 

.30509 

59 

60 

.20136 

.25214 

.21199 

.26902  i 

.22285 

.28676 

.23396 

.30541 

60 

TABLE  XIII.-VERSINES  AND  EXSECANTS. 


' 

40° 

41° 

42° 

43° 

/ 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.23396 

.30541 

.24529 

.32501 

.25086 

.3-1563 

.26865 

.36733 

c 

1 

.23414 

.30573 

.24548   .32535 

.25705 

.34599 

.26884 

.36770 

1 

2 

.2:3433 

.30605 

.24567 

.32508 

.25724 

.34634 

.26904 

.36807 

2 

3 

.23452 

.30036 

.24586 

.32002 

.25744 

.34669 

.26924 

.36844 

3 

4 

.23470 

.30668 

.24605 

.32636 

.25763 

.34704 

.26944 

.36881 

4 

5 

.23489 

.30700 

.24625 

.32669 

.25783 

.34740 

.26964 

.36919 

5 

6 

.23508 

.30732 

.24644 

.32703 

.25802 

.34775 

.26984 

.36956 

6 

7 

.23527 

.30764 

.24663 

.32737 

.25822 

.34811 

.27004 

.36993 

7 

8 

.23545 

.30796 

.24682 

.32770 

.25841 

.34846 

.27024 

.37030 

8 

9 

.23564 

.30829 

.24701 

.32804 

.25801 

.34882 

.27043 

.37'068 

9 

10 

.23583 

.30861 

.24720 

.32838 

.25880 

.34917 

.27063 

.371.05 

10 

11 

.23603 

.30893 

.24739 

.32872 

.25900 

.34953 

.27083 

.37143 

11 

12 

.28620 

.30925 

.24759 

.32905 

.25920 

.341)88 

.27103 

.37180 

12 

13 

.23G39 

.30957 

.24778 

.32939 

.259S9 

.35024 

.27123 

.87218 

13 

14 

.23658 

.30989 

.24797 

.32973 

.25959 

.35C60 

.27143 

.37255 

14 

15 

.23677 

.31022 

.24816 

.33007 

.2597'8 

.35095 

.27163 

.37293 

15 

16 

.23696 

.31054 

.24835 

.33041 

.25998 

.35131 

.27183 

.37330 

16 

17 

.23714 

.31086 

.24854 

.33075 

.26017 

.35167 

.27203 

.37368 

17 

18 

.23733 

.31119 

.24874 

.33109 

.26037 

.35203 

.27223 

.37406 

18 

19 

.28753 

.31151 

.24893 

.33143 

.26056 

.35238 

.27243 

.37443 

19 

20 

.23771 

.31183 

.24912 

.33177 

.26076 

.35274 

.27263 

.37481 

20 

21 

.23790 

.31216 

.24931 

.33211 

.26096 

.35310 

.27'2S3 

.37519 

21 

22 

.23808 

.31248 

.24950 

.33245 

.20115 

.35346 

.27303 

.37556 

22 

23 

.23827 

.31281 

.24970 

.33279 

.26135 

.35382 

.27323 

.37594 

23 

24 

.23846 

.31313 

.24989 

.33314 

.26154 

.35418 

.27343 

.37632 

24 

25 

.23865 

.31346 

.25008 

.33348 

.26174 

.35454 

.27363 

,3767'0 

25 

26 

.23884 

.31378 

.25027 

.33382 

.26194 

.35490 

.27383 

.37708 

20 

27 

.23903 

.31411 

.25047 

.33416 

.26213 

.35526 

.27403 

.37746 

27 

28 

.23922 

.81443 

.25066 

.33451 

.26233 

.35562 

.27423 

.37784 

28 

29 

.23941 

.31476 

.25065 

.33485 

.26253 

.85598 

.27443 

.37822 

29 

30 

.23959 

.31509 

.25104 

.33519 

.26272 

.35634 

.27463 

.37860 

30 

31 

.23978 

.31541 

.25124 

.33554 

.26292 

.35670 

.27483 

.37898 

31 

32 

.23997 

.31574 

.25143 

.33588 

.26312 

.35707 

.27503 

.37936 

32 

33 

.24016 

.31007 

.25162 

.33622 

.26331 

.35743 

.27523 

.37974 

33 

34 

.24035 

.31610 

.25182 

.33657 

.26351 

.35779 

.27543 

.38012 

34 

35 

.24054 

.31672 

.25201 

.33691 

.26371 

.35815 

.27563 

.38051 

85 

36 

.24073 

.31705 

.25220 

.33726 

.26390 

.35852 

.27583 

.38089 

36 

37 

.24092 

.31738 

.25240 

.33760 

.26-110 

.358B8 

.27603 

.38127 

37' 

38 

.24111 

.31771 

.25259 

.33795 

.26430 

.35924 

.27623 

.38165 

86 

39 

.24130 

.31804 

.25278 

.33830 

.20449 

.35961 

.27643 

.38204 

39 

40 

.24149 

.31837 

.25297 

.33864 

.26469 

.35997 

.27663 

.38243 

40 

41 

.24168 

.31870 

.25317 

.33899 

.26489 

.36034 

.27683 

.38280 

41 

42 

.24187 

.31903 

.25336 

.33934 

.26509 

.36070 

.277'03 

.38319 

42 

43 

.24206 

.31936 

.25356 

.33908 

.26528 

.36107 

.27723 

.38357 

43 

44 

.24:225 

.31909 

.25375 

.34003 

.26548 

.36143 

.27743 

.38396 

44 

45 

.24244 

.32002 

.25394 

.34038 

.26568 

.36180 

.27764 

.38434 

45 

46 

.24262 

.32035 

.25414 

.34073 

.26588 

.30217 

.27784 

.38473 

46 

47 

.24281 

.32068 

.25438 

.34108 

.26607 

.36253 

.27'804 

.38512 

47 

48 

.24300 

.32101 

.25452 

.34142 

.26627 

.36290 

.27824 

.38550 

48 

49 

.2-1320 

.32134 

.25472 

.31177 

.26647 

.36327 

.27844 

.38589 

49 

50 

.24339 

.32168 

.25491 

.34212 

.26667 

.36363 

.27864 

.38628 

50 

51 

.24358 

.32201 

.25511 

.34247 

.26686 

.36400 

.27884 

.38666 

51 

52 

.24377 

.32234 

,25530 

.34282 

.26706 

.36437 

.27905 

.38705 

52 

53 

.24396 

.32207 

.25549 

.34317 

.26726 

.36474 

.27925 

.38744 

53 

54 

.24415 

.32301 

.25569 

.34352 

.26746 

.36511 

.27945 

.38783 

54 

55 

.24434 

.323:34 

.25588 

.34387 

.26766 

.36548 

.27'965 

.38822 

55 

56 

.24453 

.32368 

.25608 

.34423 

.26785 

.36585 

.27985 

.38860 

56 

57 

.24472 

.32401 

.25627 

.34458 

.26805 

.36622 

.28005 

.38899 

57 

58 

.24491 

.32434 

.25647 

.34493 

.26825 

.36659 

.28026 

.38938 

58 

59 

.24:310 

.32468 

.25066 

.34528 

.26845 

.36696 

.28046 

.38977 

59 

60 

.24529 

.32501 

.25686 

.34563 

.26865 

.36733 

.28066 

.39016 

60 

331 


TABLE  XIII.— VERSINES  AND  EXSECANTS. 


4 

4° 

4 

5° 

4 

6° 

4 

7° 

Vers. 

Exsec. 

Vers. 

Exsec.  , 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.28066 

.39016 

.29289 

.41421 

.30534 

.43956 

.31800 

.46628 

0 

1 

.28086 

.39055 

.29310 

.41463 

.30)55 

.43999 

.31821 

.46674 

1 

2 

.28106 

.39095 

.29330 

.41504 

.30576 

.44042 

.31843 

.46719 

2 

3 

.28127 

.39134 

.29351 

.41545 

.30597 

.44086 

.31864 

.40765 

3 

4 

.28147 

.39173 

.29372 

.41586 

.30618 

.44129 

.31885 

.46811 

4 

5 

.28167 

.39212 

.29392 

.41627 

.30639 

.44173 

.31907 

.46857 

5 

6 

.28187 

.39251 

.29413 

.41669 

.30660 

.44217 

.31928 

.46903 

6 

7 

.28208 

.39291 

.29433 

.41710 

.30081 

.44260 

.31949 

.46949 

7 

8 

.28228 

.39330 

.29454 

.41752 

.30702 

.44304 

.31971 

.46995 

8 

9 

.28218 

.39369 

.29475 

.41793 

.30723 

.41347 

.31992 

.47041 

9 

10 

.28263 

.39409 

.29495 

.41835 

.30744 

.44391 

.32013 

.47087 

10 

11 

.28289 

.39448 

.29516 

.41876 

.30765 

.44435 

.32035 

.47134 

11 

1:2 

.28309 

.39487 

.29537 

.41918 

.30700 

.44479 

.32056 

.47180 

12 

13 

.28329 

.39527 

.29557 

.419.59 

.30807 

.44523 

.32077 

.47220 

13 

14 

.28350 

.39566 

.29578 

.42001 

.30828 

.44567 

.32090 

.47'272 

14 

15 

.28370 

.39606 

.29599 

.42042 

.30849 

.44610 

.32120 

.47310 

15 

16 

.28390 

.39646 

.29619 

.42084 

.30870 

.44654 

.32141 

.47365 

16 

17 

.28410 

.39685 

.29640 

.42126 

.30891 

.44698 

.32163 

.47411 

17 

18 

.28431 

.39725 

.29661 

.42168 

.30912 

.44742 

.32184 

.47458 

18 

19 

.28451 

.39764 

.29681 

.42210 

.30933 

.44787 

.32205 

.47501 

19 

23 

.28471 

.39804 

.29702 

.42251 

.30954 

.44831 

.32227 

.47551 

20 

21 

.28492 

.39844 

.29723 

.42293 

.30975 

.44875 

.32248 

.47598 

21 

22 

.28512 

.39884 

.29743 

.42335 

.30906 

.44919 

.32270 

.47644 

22 

23 

.28533 

.39924 

.29764 

.42377 

.31017 

.449(53 

.32291 

.47691 

23 

24 

.28553 

.39963 

.29785 

.42419 

.31038 

.45007 

.32312 

.47738 

24 

25 

.28573 

.40003 

.29805 

.42461 

.31059 

.45052 

.32334 

.47784 

25 

26 

.28503 

.40043 

.298J6 

.42503 

.31080 

.45096 

.32355 

.47831 

26 

27 

.28614 

.40083 

.298-17 

.42545 

.31101 

.45141 

.32377 

.47878 

27 

28 

.28634 

.40123 

.29868 

.42587 

.31122 

.45185 

.32398 

.47925 

28 

29 

.28655 

.40163 

.29888 

.42630 

.31113 

.45229 

.32420 

.47972 

29 

30 

.28675 

.40203 

.29909 

.42672 

.31165 

.45274 

.32441 

.48019 

30 

31 

.28605 

.40243 

.29930 

.42714 

.31186 

.45319 

.32462 

.48066 

31 

32 

.28716 

.40283 

.29951 

.42756 

.31207 

.45363 

.32484 

.48113 

32 

33 

.28736 

.40324 

.29971 

.42799 

.31228 

.45408 

.82505 

.48160 

33 

34 

.28757 

.40364 

.29992 

.42841 

.31249 

.45452 

.32527 

.48207 

34 

35 

.28777 

.40404 

.80013 

.42883 

.31270 

.45497 

.32548 

.48254 

35 

36 

.28797 

.40444 

.30034 

.42926 

.31291 

.45542 

.32570 

.48301 

36 

37 

.28818 

.40485 

.30054 

.42968 

.31312 

.45587 

.32591 

.48349 

37 

38 

.28838 

.40525 

.30075 

.43011 

.31334 

.45631 

.32613 

.48396 

38 

39 

.28859 

.40565 

.30096 

.43053 

.313r,5 

.45678 

.32634 

.48443 

39 

40 

.28879 

.40606 

.30117 

.43096 

.31376 

.45721 

.32656 

.48491 

40 

41 

.28900 

.40646 

.30138 

.43139 

.31397 

.45766 

!  .32677 

.48538 

41 

42 

.28920 

.40087 

.30158 

.43181 

.31418 

.45811 

.32(5:19 

148586 

42 

43 

.28941 

.40727 

.30179 

.43224 

.31439 

.45856 

.82720 

.48633 

43 

44 

.28961 

.40768 

.30200 

.432(57 

.31461 

.45901 

.30712 

.48(581 

44 

45 

.28981 

.40808 

.30221 

.43310 

.31482 

.45046 

.32763 

.48728 

45 

46 

.29002 

.40849 

.30242 

.43352 

.31503 

.45992 

.32785 

.48776 

46 

47 

.29022 

.40890 

.30263 

.48896 

.31524 

.46037 

.32806 

.48824 

47 

48 

.29043 

.40930 

.30283 

.43438 

.31515 

.46082 

.32828 

.48871 

48 

49 

.29063 

.40971 

.30304 

.43481 

.315117 

.46127 

.32849 

.480  10 

49 

50 

.29084 

.41012 

.30325 

.43524 

.31588 

.46173 

.32871 

.48967 

50 

51 

.29104 

.41053 

.30346 

.43507 

.31609 

.46218 

.32893 

.49015 

51 

52 

.29125 

.41093 

.30367 

.43610 

.31630 

.46263 

.32914 

.490(53 

52 

53 

.29145 

.41134 

.30388 

.43653 

.31651 

.46309 

.32936 

.49111 

53 

54 

.29166 

.41175 

.30409 

.43696 

.31673 

.40354 

.32957 

.49159 

54 

55 

.29187 

.41216 

.30430 

.437':!9 

.31694 

.46-400 

.32979 

.40207 

55 

56 

.29207 

.41257 

.30451 

.43783 

.31715 

.46445 

.83001 

.49255 

56 

57 

.29228 

.41298 

.30471 

.43826 

.31736 

.46491 

.33022 

.49308 

57 

'-58 

.2:);.'  18 

.41888 

.30102 

.43869 

.31758 

.46537 

.33044 

.49351 

58 

59 

.29209 

.41380 

.30513 

.43012 

.31779 

.46582 

.33065 

.49399 

59 

60 

.29289 

,41421 

.30534 

.43956 

.31800 

.46628 

.33087 

.49448 

60 

333 


TABLE  XIII.-VERSINES  AND  EXSECANTS. 


t 

48° 

49° 

50° 

51° 

f 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.33087 

.49448 

!  .34394 

.52425 

.35721 

.55572 

.37068 

.58902 

0 

1 

.33109 

.49496 

,34416 

.52476 

.35744 

.55626 

.37091 

.58959 

1 

2 

.33130 

.49544 

.34438 

.52527 

.35766 

.55680 

.37113 

.59016 

2 

.33152 

.49593 

.34460 

.52579 

.35788 

.55734 

.37136 

.59073 

3 

4 

.&3173 

.49641 

.34482 

.52630 

.35810 

.55789 

.37158 

.59130 

4 

5 

.33195 

.49690 

.34504 

.52681 

.35833 

.55843 

.37181 

.59188 

5 

6 

.33217 

.49738 

.34526 

.52732 

.35855 

.55897 

.37204 

.59245 

6 

7 

.33238 

.49787 

.34548 

.52784 

.35877 

.55951 

.37226 

.59302 

7 

8 

.33260 

.49835 

.34570 

.52835 

.35900 

.56005 

.37249 

.59360 

8 

9 

.33282 

.49884 

.34592 

.52886 

.35922 

.56060 

.37272 

.59418 

9 

10 

.33303 

.49933 

.34614 

.52938 

.35944 

.56114 

.37294 

.59475 

10 

11 

.33325 

.49981 

.34636 

.52989 

.35967 

.56169 

.37317 

.59533 

11 

12 

.33347 

.50030 

.34658 

.53041 

.35989 

.56223 

.37340 

.59590 

12 

13 

.33368 

.5007'9 

.34680 

.53092 

.36011 

.56278 

.37362 

.5964H 

13 

14 

.33390 

.50128 

.34702 

.53144 

.36031 

-56332 

.37385 

.59700 

14 

15 

.33412 

.50177 

.34724 

.53196 

.36056 

.56387 

.37408 

.59764 

15 

16 

.33434 

.50226 

.34746 

.53247 

.36078 

.56442 

.37430 

.59822 

16 

17 

.a3455 

.50275 

.34768 

.53299 

.36101 

.56497 

.37453 

.59880 

17 

18 

.33477 

.50324 

.34790 

.53351 

.36123 

.56551 

.37476 

.59938 

18 

19 

.33499 

.50373 

.34812 

.53403 

.36146 

.56606 

.37498 

.59996 

19 

20 

.33520 

.50422 

.34834 

.53455 

.36168 

.56661 

.37521 

.60054 

20 

21 

.33542 

.50471 

.34856 

.53507 

.36190 

.56716 

.37544 

.60112 

21 

22 

.33564 

.50521 

.34878 

.53559 

.36213 

.56771 

.87567 

.60171 

22 

23 

.33586 

.50570 

.34900 

.53611 

.36235 

.56826 

.37589 

.60229 

23 

24 

.33607 

.50619 

.34923 

.53663 

.36258 

.56881 

.37612 

.60287 

24 

25 

.33829 

.50669 

.34945 

.53715 

.36280 

.56937 

.37635 

.60346 

25 

26 

.33651 

.50718 

.34967 

.53768 

.36302 

.56992 

.37658 

.60404 

26 

27 

.33673 

.50767 

.34989 

.53820 

.36325 

.57047 

.37680 

.60463 

27 

28 

.33694 

.50817 

.35011 

.53872 

.36347 

.57103 

.37703 

.60521 

28 

29 

.33716 

.50866 

.35033 

.53924 

.36370 

.57158 

.37726 

.60580 

29 

30 

.33738 

.50916 

.35055 

.53977 

.36392 

.57213 

.37749 

.60639 

30 

31 

.33760 

.50966 

.35077 

.54029 

.36415 

.57269 

.37771 

.60698 

31 

32 

.33782 

.51015 

.35099 

.54082 

.36437 

.57324 

.37794 

.60756 

32 

33 

.33803 

.51065 

.35122 

.54134 

.36460 

.57380 

.37817 

.60815 

33 

34 

.33825 

.51115 

.35144 

.54187 

.36482 

.57436 

.37840 

.60874 

34 

35 

.33847 

.51165 

.35166 

.54240 

.36504 

.57491 

.37862 

.60933 

35 

36 

.33869 

.51215 

.35188 

.54292 

.36527 

.57547 

.37885 

.60992 

36 

37 

.33891 

.51265 

.35210 

.54345 

.36549 

.57603 

.37908 

.61051 

37 

38 

.33912 

.51314 

.35232 

.54398 

.36572 

.57659 

.37931 

.61111 

38 

39 

.33934 

.51364 

.35254 

.54451 

.36594 

.57715 

.37954 

.61170 

39 

40 

.33956 

.51415 

.35277 

.54504 

.36617 

.57771 

.37976 

.61229 

40 

41 

.33978 

.51465 

.35299 

.54557 

.36639 

.57827 

.37999 

.61288 

41 

42   .34000 

.51515 

.35321 

.54610 

.36662 

.57883 

.38022 

.61348 

42 

43   .84022 

.51565 

.35343 

.54663 

.36684 

.57939 

.38045 

.61407 

43 

44  i  .340-14 

.51615 

.35365 

.54716 

.36707 

.57995 

.38068 

.61467 

44 

45  i  .34005 

.51665 

.35388 

.54769 

.36729 

.58051 

.38091 

.61526 

45 

46  !  .34087 

.51716 

.35410 

.54822 

.36752 

.58108 

.38113 

.61586 

46 

47   .34109 

.51766 

.35432 

.54876 

.36775 

'.58164 

.38136 

.61646 

47 

48 

.34131 

.51817 

.35454 

.54929 

.36797 

.58221 

.38159 

.61705 

48 

49 

.34153 

.51867 

.35476 

.54982 

.36820 

.58277 

.38182 

.61765 

49 

50 

.34175 

.51918 

.35499 

.55036 

.36842 

.58333 

.38205 

.61825 

50 

51 

.34197 

.51968 

.35521 

.55089 

.36865 

.58390 

.38228 

.61885 

51 

52 

.34219 

.52019 

.35543 

.55143 

.36887 

.58447 

.38251 

.61945 

52 

53 

.34241 

.52069 

'  .35565 

.55196 

.36910 

.58503 

.38274 

.62005 

53 

54 

.34262 

.52120 

.35588 

.55250 

.36932 

.58560  I 

.38296 

.62065 

54 

55 

.34284 

.52171 

.35610 

.55303 

.36955 

.58617  i 

.38319 

.62125 

55 

56 

.34306 

.52222 

.35632 

.55357 

.3697'8 

.58674  ! 

.38342 

.62185 

56 

57 

.34328 

.52273 

.35654 

.55411 

.S7000 

.58731  i 

.38365 

.62246 

57 

58 

.34.350 

.52323 

.35677 

.55465  i  .&r003 

.58788 

.38388 

.62306 

58 

59 

.34372 

.52374 

.35699 

.55518   .37015 

.58845  | 

.38411 

.62366 

59 

60 

.34394 

.52425 

.35721 

.55572  1  .37068 

.58902  1 

.38434 

.62427 

60 

TABLE  XIIL— VERSINES  AND  EXSECANTS. 


5 

2° 

5 

3° 

5 

i° 

5 

5° 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.38434 

.62427 

.39819 

.66164 

.41221 

.70130 

42042 

.74345 

0 

1 

.38457 

.62487 

.39842 

.66228 

.41245 

.70198  | 

.42600 

.74417 

1 

2 

.38480 

.62548 

.89665 

.66292 

.41209 

.70207  I 

.42090 

.74490 

2 

3 

.38503 

.62609 

.39888 

.66357 

.41292 

.70,335 

.42714 

.74562 

3 

4 

.38526 

.62669 

.39911 

.66421 

.41316 

.70403  ; 

.42738 

.74035 

4 

5 

.38549 

.62730 

.39935 

.60486 

.41339 

.70472 

.42762 

.74708 

5 

6 

.38571 

.62791 

.39958 

.66550 

.41303 

.70540 

.42785 

.74781 

6 

7 

.38594 

.62852 

.39981 

.66615 

.41386 

.70609 

.42809 

.74a54 

7 

8 

.38617 

.62913 

.40005 

.66679 

.41410 

.70677 

.42833 

.74927 

8 

9 

.38640 

.62974 

.40028 

.66744 

.41433 

.70746 

.42857 

.75000 

9 

10 

.38663 

.63035 

.40051 

.60809 

.41457 

.70815 

.42881 

.75073 

10 

11 

.38686 

.63096 

.40074 

.66873 

.41481 

.70884 

.42905 

.75146 

11 

12 

.38709 

.63157 

.40098 

.66938 

.41504 

.70953 

.42929 

.75219 

12 

13 

.38732 

.63218 

.40121 

.67003 

.41528 

.71022 

.42953 

.75293 

13 

14 

.38755 

.63279 

.40144 

.67'068 

.41551 

,71091 

.42970 

.75366 

14 

15 

.38778 

.  63341 

.40168 

.67133 

.41575 

.71100 

.43000 

.75440 

15 

16 

.38801 

.63402 

.40191 

.67199 

.41599 

.71229 

.43024 

.75513 

16 

17 

.38824 

.63464 

.40214 

.67264 

.41622 

.71298 

.43048 

.75587 

17 

18 

.38847 

.63525 

.40237 

.67329 

.41646 

.71368  i 

.43072 

.75661 

18 

19 

.38870 

.63587 

.40261 

.07394 

.41670 

.71437  i 

.43096 

.75734 

19 

20 

.38893 

.63648 

.40284 

.67460 

.41693 

.71506  ; 

.43120 

.75808 

20 

21 

.38916 

.63710 

.40307 

.67525 

.41717 

.71576  ] 

.43144 

.75882 

21 

22 

.38939 

.63772 

.40331 

.67591 

.41740 

.71040 

.43168 

.75956 

22 

23 

.38962 

.63834  i 

.40354 

.67656 

.41764 

.71715 

.43192 

.76031 

23 

24 

.38985 

.63895  I 

.40378 

.67722 

.41788 

.71785  : 

.43216 

.76105 

24 

25 

.39009 

.63957  ! 

.40401 

.67788 

.41811 

.71855  i 

.43240 

.76179 

25 

26 

.39032 

.64019  ; 

.40424 

.67853 

.41835 

.71925  I 

.43204 

.76253 

26 

27 

.39055 

.64081 

.40448 

.67919 

.41859 

.71995 

.43287 

.76328 

27 

28 

.39078 

.64144 

.40471 

.67985 

.41882 

.72065 

.43311 

.76402 

28 

29 

.39101 

.64206 

.40494 

.68051 

.41906 

.72135 

.43335 

.76477 

29 

30 

.39124 

.64268 

.40518 

.68117 

.41930 

.72205 

.43359 

.76552 

30 

31 

.39147 

.64330 

.40541 

.68183 

.41953 

.72275 

.43383 

.76626 

31 

32 

.39170 

.64393 

.40565 

.68250 

.41977 

.72346  i 

.43407 

.76701 

32 

33 

.39193 

.64455 

.40588 

.68316 

.42001 

.72416 

.43431 

.76776 

33 

34 

.39216 

.64518 

.40611 

.68382 

.42024 

.72487 

.43455 

.76851 

34 

a5 

.39239 

.64580 

.40635 

.68449 

.42048 

.72557 

.43479 

.76926 

35 

36 

.39262 

.64643 

.40658 

.68515 

.42072 

.72628 

.43503 

.77001 

36 

37 

.39286 

.64705 

.40682 

.68582 

.42096 

.72698 

.43527 

.77077 

37 

38 

.39309 

.64768 

.40705 

.68648 

.42119 

.72769 

.43551 

.77152 

38 

89 

.39332 

.64831 

.40728 

.68715 

.42143 

.72840 

.43575 

.77227 

39 

40 

.39355 

.64894 

.40752 

•  .687'82 

.42107 

.72911 

.43599 

.77303 

40 

41 

.39378 

.64957 

.40775 

.68848 

.42191 

.72982 

.43623 

.77378 

41 

42 

.39401 

.65020 

.40799 

.68915 

.42214 

.7-3053 

.43047 

.77454 

42 

43 

.39424 

.65083 

.40822 

.68982 

.42238 

.73124 

.43671 

.77530 

43 

44 

.39447 

.65146 

.40846 

.69049 

.42262 

.73195 

.43695 

.77606 

44 

45 

.39471 

.65209 

.40809 

.09116 

.42285 

.73267 

.43720 

.77'681 

45 

46 

.39494 

.65272 

.40893 

.09183 

.42309 

.73338 

.43744 

.77757 

46 

47 

.39517 

.G5336 

.40916 

.09250 

.42333 

.73409 

.43708 

.77833 

47 

48 

.39540 

.65399  i 

.40939 

.09318 

.42357 

.73481  , 

.43792 

.77910 

48 

49 

.39563 

.65462 

.40963 

.69385 

.42381 

.78552 

.43816 

.77986 

49 

50 

.39586 

.65526 

.40986 

.69452 

.42404 

.73634 

.43840 

.78062 

50 

51 

.39610 

.65589 

.41010 

.69520 

.42428 

.73696 

.43864 

.78138 

51 

52 

.39633 

.65653 

.41033 

.69587 

.42452 

.78768 

.43888 

.78215 

52 

53 

.39056 

.65717 

.41057 

.69655 

.424:0 

.7'3840  , 

.43912 

.78291 

53 

54 

.39679 

.65780 

.4108Q 

.69723 

.42199 

.73911 

.43936 

.78308 

54 

55 

.39702 

.65844 

.41104 

.69790 

.42523 

178988 

.43900 

.78445 

55 

56 

.39726 

.65908 

.41127 

.69858 

.42547 

74056 

.43984 

.78521 

56 

57 

.39749 

.65972 

.41151 

.69926 

.42571 

.74128 

.nons 

.78f)!)8 

57 

'-58 

.39772 

.66036 

.41174 

.69994 

.42505 

.74200 

.44032 

.78075 

58 

59 

.39795 

.66100 

.41198 

.70063 

.42019 

.74272 

.44057 

.78752 

59 

60 

.39819 

.66164 

.41221 

.70130 

.42642 

.74345 

.44081 

.78829 

60 

334 


TABLE  XIII.— VERSINES  AND  EXSECANTS. 


5 

6° 

5 

7° 

5 

>° 

5 

|? 

/ 

Vers. 

Exsec. 

Vers. 

Exsec. 

i  Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.44081 

.78829 

.45536 

.83608 

.47008 

.88708 

.48496 

.94160 

0 

1 

.44105 

.78906 

.45560 

.83690 

.47033 

.88796 

.48521 

.94254 

1 

2 

.44129 

.78984 

.45585 

.83773 

.47057 

.88884 

.48546 

.94:349 

2 

3 

.44153 

.79061 

.45609 

.83855 

.47082 

.88972 

i  .48571 

.94443 

3 

4 

.44177 

.79138 

.45634 

.83938 

.47107 

.89060 

.48596 

.94537 

4 

5 

.44201 

.79216 

.45658 

.84020 

.47131 

.89148 

.48621 

.94632 

5 

6 

.44225 

.79293 

.45683 

.84103 

.47156 

.89237 

.48646 

.94726 

6 

7 

.44250 

.79371 

.45707 

.84186 

.47181 

.89325 

.48671 

.94821 

7 

8 

.44274 

.79449 

.45731 

.84269 

.47206 

.89414 

.48696 

.94916 

8 

9 

.44295 

.79527 

.45756 

.84352 

.472:30 

.89503 

.48721 

.95011 

9 

10 

.44322 

.79604 

.45780 

.84435 

.47255 

.89591 

.48746 

.95106 

10 

11 

.44346 

.79682 

.45805 

.84518 

.47280 

.89680 

.48771 

.95201 

11 

12 

.44370 

.79761 

.45829 

.84601 

.47304 

.89769 

.48796 

.95296 

12 

13 

.44395 

.79839 

.45854 

.84685 

.47329 

.89858 

.48821 

.95392 

13 

14 

.44419 

,  .79917 

.45878 

.84768 

.47354 

.89948 

.48846 

.95487 

14 

15 

.44443 

.79995 

.45903 

.84852 

.47379 

.90037 

.48871 

.95583 

15 

16 

.44467 

.80074 

.45927 

.84935 

.47403 

.90126 

.48896 

.95678 

16 

17 

.44491 

.80152 

.45951 

.85019 

.47428 

.90216 

.48921 

.95774 

17 

18 

.44516 

.80231 

.45976 

.85103 

.47453 

.90305 

.48946 

.95870 

18 

19 

.44540 

.80309 

.46000 

.85187 

.47478 

.90395 

.48971 

.95966 

19 

20 

.44564 

.80388 

.46025 

.85271 

.47502 

.90485 

.48996 

.96062 

20 

21 

.44588 

.80467 

.46049 

.85355 

.47527 

.90575 

.49021 

.96158 

21 

oo 

.44612 

.80546 

.46074 

.85439 

.47552 

.90665 

.49046 

.96255 

22 

23 

.44637 

.80625 

.46098 

.85523 

.47577 

.90755 

.49071 

.96351 

23 

24 

.44661 

.80704 

.46123 

.85608 

.47601 

.90845 

.49096 

.96448 

24 

25 

.446S5 

.80783 

.46147 

.85692 

.47626 

.90935 

.49121 

.96544 

25 

26 

.44709 

.80862 

.46172 

.85777 

.47651 

.91026 

.49146 

.96641 

26 

27 

.44734 

.80942 

.46196 

.85861 

.47676 

.91116 

.49171 

.967'38 

27 

28 

.44758 

.81021 

.46221 

.85946 

.47701 

.91207 

.49196 

.96835 

28 

29 

.44782 

.81101 

.46246 

.S6031 

.47725 

.91297 

.49221 

.96932 

29 

30 

.44806 

.81180 

.46270 

.86116 

.47750 

.91388 

.49246 

.97029 

30 

31 

.44831 

.81260 

.46295 

.86201 

.47775 

.91479 

.49271 

.97127 

31 

32 

.44855 

.81340 

.46319 

.86286 

.47800 

.91570 

.49296 

.97224 

33 

33 

.44879 

.81419 

.46344 

.86371 

.47825 

.91661 

.49321 

.97322 

33 

34 

.44903 

.81499 

.46368 

.86457 

.47-849 

.91733 

.49346 

.97420 

34 

85 

.44928 

.81579 

.46393 

.86542 

.47874 

.91844 

.49372 

.97517 

35 

36 

.44952 

.81659 

.46417 

.86627 

.47899 

.91935 

.49397 

.97-615 

36 

37 

.44976 

.81740 

.46442 

.86713 

.47924 

.92027 

.49422 

.97713 

37 

38 

.45001 

.81820 

.46466 

.86799 

.47949 

.92118 

.49447 

.97811 

38 

39 

.45025 

.81900 

.46491 

.86885 

.47974 

.92210 

.49472 

.97910 

39 

40 

.45049 

.81981 

.46516 

.86990 

.47998 

.92302 

.49497 

.98008 

40 

41 

.45073 

.82061 

.46540 

.87056 

.48023 

.92394 

.49522 

.98107 

41 

42 

.45098 

.82142 

.46505 

.87142 

.48048 

.92486 

.49547 

.98205 

42 

43 

.45122 

.82222 

.46589 

.87229 

.48073 

.92578 

.49572 

.98304 

43 

44 

.45146 

.82303 

.46614 

.87315 

.48098 

.92670 

!  .49597 

.98403 

44 

45 

.45171 

.82384 

.46639 

.87401 

.48123 

.92762 

1  .49623 

.98502 

45 

46 

.45195 

.82465 

.46663 

.87488  ! 

.48148 

.92855 

i  .49648 

.98601 

46 

47 

.45219 

.82546 

.46688 

.87574 

.48173 

-  .92947 

.49673 

.98700 

47 

48 

.45244 

.82627 

.46712 

.87661 

.48197 

.93040 

.49698 

.98799 

48 

49 

.45268 

.82709 

.46737 

.87748 

.48222 

.93133 

.49723 

.98899 

49 

50 

.45292 

.82790 

.46762 

.87834 

.48247 

.93226 

.49748 

.98998 

50 

51 

.45317 

.82871 

.46786 

.87921 

.48272 

.93319 

.49773 

.99098 

51 

52 

.45341 

.82953 

.46811 

.88008 

.48207 

.93412 

.49799 

.99198 

52 

53 

.45365 

.83034 

.46836 

.88095 

.48322 

.  93505 

.49824 

.99298 

53 

54 

.45390 

.83116 

.46860 

.88183 

.48347 

.93598 

.49849 

.99398 

54 

•55 

.45414 

.83198 

.46885 

.88270 

.48373 

.93692 

.49874 

.99498 

55 

56 

.45439 

.83280 

.46909 

.88357  ! 

.48396 

.93785 

.49899 

.99598 

56 

57 

.45463 

.83362 

.46934 

.88445  : 

.48421 

.93879 

.49924 

.99698 

57 

58 

.45487 

.83444 

.46959 

.88532 

.48446 

.93973 

.49950 

.99799 

58 

59 

.45512 

.83526 

.46983 

.8K020 

.48471 

.94066 

.49975 

.99899 

59 

60 

.45536 

.83608 

.47008 

.88708 

.48496 

.94160 

.50000 

.00000 

60 

335 


TABLE  XIII.-VERSINES  AND  EXSECANTS. 


/ 

60°           ||           61°                       62° 

63° 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

( 

.50000 

1.00000 

1     .51519 

1.06267 

.53053      1.13005 

.54601 

1.20269      0 

; 

.50025 

1.00101 

.51544 

1.06375 

.53079 

1.13122 

.54627 

1.20395      1 

j 

.50050 

1.00202 

.51570     1.06483 

.53104 

1  .  13239 

.54653 

1.20521 

2 

j 

.50076      1.00303 

.51595      1.0G5JW 

.53130 

1.13356 

.54679 

1.20647 

J 

t 

.50101      1.00404 

.51621      1.06701 

.53156 

1.13473 

.54705      1.20773 

4 

5 

.50126 

1.00505 

.51646      1.06809 

.53181 

1.13590 

.54731      1.20900 

G 

.50151 

1.00607 

.51672 

1.06918 

.53207 

1.13707 

.54757 

1.21026 

t 

r 

.50176 

1.00708 

.51697 

1.07027 

.53233 

1.13825 

.547'82 

1.21153 

ij 

8 

.50202 

1.00810 

i    .51723 

1.07137 

.53258 

1.13942 

!   .54808     1.21280 

8 

( 

.50227 

1.0091-2 

.51748      1.07246 

.53284 

1.14060 

[   .54834 

1.21407 

1C 

.50252 

1.01014 

i    .51774 

1.07356 

.53310 

1.14178 

.54860 

1.21535 

1C 

11 

.50277 

1.01116 

.51799 

1.07465 

.53-336 

1.14296 

.54886 

1.21662 

11 

12 

.50303 

1.01218 

.51825 

1.07575 

.53361 

1.14414 

.54912 

1.21790 

12 

13 

.50328 

1.01320 

.51850  i  1.07685 

.53387 

1.14533 

.54938 

1.21918 

14 

.50353 

1.01422 

.51876 

1.07795 

.53413 

1.14651 

1   .54964 

1.22045 

14 

15 

.50378 

1.01525 

.51901 

1.07905 

.53489 

1.14770 

.54990 

1.22174 

15 

16 

.50404 

1.01628 

.51927 

1.08015 

.53464 

1.14889 

.550113 

1.22302 

16 

17 

.50429 

1.01730 

.51952 

1.08126 

.53490 

1.15008 

.55042 

1.22430    17 

18 

.50454 

1.01833 

.51978 

1.08236 

.53516 

1.15127 

.55068 

1.225:.!) 

18 

19 

.50479 

1.01936 

.52003 

1.08347 

.53542 

1.15246 

.55094 

1.22688 

19 

20 

.50505 

1.02039 

.52029 

1.08458 

.53507 

1.15306 

.55120 

1.2-2817 

20 

21 

.50530 

1.02143 

.520.54 

1.08569 

.53503 

1.15485 

.55146 

1.22946 

21 

22 

.50555 

1.02246 

.52080 

1.08680 

.58619 

1.15605 

.55172 

1.23075 

22 

23 

.50581 

1.02349 

.52105 

1.08791 

.58645 

1.15725 

.55198 

1.23205 

O'.> 

24 

.50606 

1.02453 

.52131 

1.08903 

.53670 

1.15845 

.55224 

1.23334 

24 

25 

.50631 

1.02557 

.52156 

1.09014 

.53696 

1.15965 

.55250 

L28464 

25 

26 

.50656 

1.02661 

.52182 

1.09126 

.53722 

1.16085 

.55276 

1.28594 

2(5 

27 

.50682 

1.02765 

.52207 

1.09238 

.53748 

1.16208 

.55302 

1.28724 

27 

28 

.50707 

1.02869 

.52233 

1.09350 

.53774 

1.16326 

.55328 

l.£K">  •!>« 

29 

.50732 

1.02973 

.52259 

1.09462 

.53799 

1.16447 

.55354 

1.23985    29 

30 

.50758 

1.03077 

.52284 

1.09574 

.53825 

1.16568 

.55380 

1.24116 

30 

31 

.50783 

1.03182 

.52310 

1.09686 

.53851 

1.16689 

.55406 

1.24247 

31 

32 

.50808 

1.03286 

.52335 

1.09799 

.53877 

1.16810 

.55432 

1.24378 

32 

33 

.50834 

1.03391 

.52361 

1.09911 

.53903 

1.16932 

.55458 

1.24509  133 

34 

.50859 

1.03496 

.52386 

1.10024 

.53928 

1.17053 

.55484 

1.2415-10    :-Jt 

35 

.50884 

1.03601 

.52412 

1.10137 

.53954 

1.17175 

.55510 

1.2477'2    35 

36 

.50910 

1.03706 

.52438 

1.10250 

.53980 

1.17297 

.55536 

1.2491  .3    36 

37 

.50935 

1.03811 

.52463 

1.10363 

.54006 

1.17419 

.55563 

1.25035    37 

38 

.50960 

1.03916 

.52489 

1.10477 

.54032 

1.17541 

.55589 

1  25167  !38 

30 

.50986 

1.04022 

.52514 

1.10590 

.54058 

1.17663 

.55615 

l.Ji.VJOO  !39 

40 

.51011 

1.04128 

.52540 

1.10704 

.54083 

1.17786 

.55641 

1.S543S 

40 

41 

.51036 

1.04233 

.52566 

1.10817 

.54109 

1.17909 

.55667 

1.25505 

41 

42 

.51062 

1.04339 

.52591 

1.10931 

.54135 

1.18031 

.55693 

1.26697 

42 

43 

.51087  i  1.04445 

.52617 

1.11045 

.54161 

1.18154 

.55719 

1.25830 

43 

44 

.51113 

1.04551 

.52642 

1.11159 

.54187 

1.18277 

.55745 

1.26968 

44 

45 

.51138 

1.04658 

.52668 

1.11274 

.54213 

1.18401 

.  55771 

1.26097  |45 

46 

.51163 

1.04764 

.52694 

1.11388  ; 

.54238 

1.18524 

.55797 

1.26230  1  46 

47 

.51189 

1.0487'0 

.52719 

1.11503 

.54264 

1.18648 

.55823 

1.26364    47 

48 

.51214 

1.04977 

.52745 

1.11617    i   .54290 

1.18772 

.55849 

1.26498  1  48 

49 

.51239 

1.05084 

.52771 

1.11732 

.54316 

1.18895 

.55876 

1.2»i(W2    49 

50 

.51265 

1.05191 

.52796 

1.11847 

.54342 

1.19019 

.55902 

1.26766 

50 

51 

.54290 

1.05298 

.52822 

1.11963 

.54368 

1.19144 

.55928 

1.26900 

51 

52 

.51316 

1.05405 

.52&J8 

1.12078  ! 

.54394 

1.19268 

.55954 

1.  271)85    52 

53 

.51341 

1.05512 

.52878 

1.12193 

.54420 

1.WJ93 

.55980 

1.27169    53 

54 

.51366 

1.05619 

.52899 

1.12309 

.54446 

1.19517 

.56006 

1.27304    54 

55 

.51392 

1.05727 

.52924 

1.12425 

.54471 

1.19648 

.56032  i   1.274:1!)    55 

56 

.51417 

1.05a35 

.52950 

1.12540 

54497 

1.19767 

.56058 

1.27574    r><; 

>7 

.51443 

1.05942 

.5297'6 

1.12657       .54523 

1.19892 

.56084 

1.27710    57 

58 

.51468 

1.06050 

.58001 

1.1  277-:!       .54549 

1.20018 

.56111 

1.27845  |58 

59 

.51494 

1.06158 

.53027 

1.12889       .54575 

1.20143 

.56137 

1.27981 

59 

60 

.51519  1  1.06267 

.53053 

1.13005       .54001 

1.20269 

.56163 

1.28117 

60 

336 


TABLE  XIII.— VERSINES  AND  EXSECANTS. 


6 

4° 

6 

5° 

6 

6° 

6 

7° 

Vers. 

Exsec.  j 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.56163 

1.28117 

.57738 

1.36620  ' 

.59326 

1.45859 

.60927 

1.55930 

0 

1 

.56189 

1.28253 

.57765 

1.36768  r 

.59353 

1.46020 

.60954 

1.56106 

1 

'.) 

.56215 

1.28390 

.57791 

1.36916 

.59379 

1.46181 

.60980 

1.56282 

2 

3 

.56241 

1.28526 

.57817 

1.37064 

.59406 

1.46342 

.61007 

1.56458 

3 

4 

.56267 

128663 

.57844 

1.37212 

.59433 

1.46504 

.61034 

1.56634 

4 

5 

.56294 

.28800 

.57870 

1.37361 

.59459 

1.46665 

.61061 

1.56811 

5 

6 

.50*30 

.28937  i 

.57896 

1.37509  i 

.59486 

1.46827 

.61088 

1.56988 

6 

; 

.56346 

.29074  i 

.57923 

1.37658  ! 

.59512 

1.46989 

.61114 

1.57165 

7 

8 

.56372 

1.29211  i 

.57949 

1.37808  i 

.59539 

1.47152 

.61141 

1.57342 

8 

9 

.56398 

.29349  i 

.57976 

1.37957  ! 

.59566 

1.47314 

.61168 

1.57520 

9 

10 

.56425 

1  .29487  i 

.58002 

1.38107  i 

.59592 

1.47477 

.61195 

1  .57698 

10 

11 

.56451 

1.29625  i 

.58028 

1.38256 

.59619 

1.47640 

.61222 

1.57876 

11 

12 

.56477 

1-29763  1 

.58055 

1.38406 

.59645 

1.47804 

.61248 

1.58054 

12 

13 

.56503 

1.29901  i 

.58081 

1.38556 

.59672 

1.47967 

.61275 

1.58233 

13 

14 

.56529 

1.30040  ! 

.58108 

1.38707 

.59699 

1.48131 

.61302 

1.58412 

14 

15 

.56555 

1.30179  : 

.58134 

1.38857 

.59725 

1.48295 

.61329 

1.58591 

15 

1C, 

.56582 

1.30318 

.58160 

1.39008 

.59752 

1.48459 

.61356 

1.58771 

16 

17 

.56608 

1.30457 

.58187 

1.39159 

.59779 

1.48624 

.61383 

1.58950 

17 

18 

.5(5(534 

.30596  i 

.58213 

1.39311 

.59805 

1.48789 

.61409 

1.59130 

18 

19 

.566(50 

.30735  ! 

.58240 

1.39462 

.59832 

1.48954 

.61436 

1.59311 

19 

20 

.56687 

.30875 

.58266 

1.39614  ! 

.59859 

1.49119 

.61463 

1.59491 

20 

21 

.56713 

.31015 

.58293 

1.39766 

.59885 

1.49284 

.61490 

1.59672 

21 

-.'•2 

.50739 

.31155 

.58319 

1.39918 

.59912 

1.49450 

.61517 

1.59853 

22 

23 

.56765 

.31295  ! 

.583-15 

1.40070 

.59938 

1.49616 

.61544 

1.60035 

23 

24 

.56791 

.31436  i 

.58372 

1.40222 

.59965 

1.49782 

.61570 

1.60217 

24 

25 

.5(5818 

.31576  ! 

.58398 

1.40375 

.59992 

1.49948  1 

.61597 

1.60399 

25 

-,><; 

156844 

.31717 

.58425 

1.40528 

.60018 

1.50115 

.61624 

1.60581 

26 

27 

.515870 

.31858  i 

.58451 

1.40681 

.60045 

1.50282 

.61651 

1.60763 

27 

•2S 

.56896 

.31999 

.58478 

1.40835 

.60072 

1.50449 

.61678 

1.60946 

28 

29 

.56923 

.32140 

.58504 

1.40988 

.60098 

1.50617 

.61705 

1.61129 

29 

30 

.56949 

.32282 

.58531 

1.41142 

.60125 

1.50784 

.61732 

1.61313 

30 

31 

.56975 

.32424 

.58557 

1.41296 

.60152 

1.50952 

.61759 

1.61496 

31 

8-2 

.57001 

.32566 

.58584 

1.41450 

.60178 

1.51120 

.61785 

1.61680 

32 

33 

.57028 

.32708 

.58610 

1.41605 

.60205 

1.51289 

.61812 

1.61864 

33 

34 

.57054 

.32850  1 

.58637 

1.41760 

.60232 

1.51457 

.61839 

1.62049 

34 

35 

.57080 

.32993  ! 

.58663 

1.41914 

.60259 

1.51626 

.61866 

1.62234 

35 

30 

.57106 

.33135 

.58690 

1.42070 

.60285 

1.51795 

.61893 

1.62419 

3(5 

37 

.57133 

.33278 

.58716 

1.42225 

.60312 

1.51965 

.61920 

1.62604 

37 

88 

.57159 

.33422 

.58743 

1.42380 

.60339 

1.52134 

.61947 

1.62790 

88 

39 

.57185 

.33565  I 

.58769 

1.42536 

.60365 

1.52304 

.61974 

'1.62976 

39 

40 

.57212 

.33708 

.58796 

1.42692 

.60392 

1.52474 

.62001 

1.63162 

40 

11 

.57288 

.33852 

.58822 

1.42848 

.60419 

1.52645 

.62027 

1.63348 

41 

i;2 

.57264 

!  33996 

.58849 

1.43005 

.60445 

1.52815  ! 

.62054 

1.63535 

4-2 

48 

.57201 

.34140 

.58875 

1.43162 

.60472 

1.52986 

.62081 

1.63722 

43 

44 

.57317 

.34284  ; 

.58902 

1.43318 

.60499 

1.53157 

.62108 

1.63909 

44 

15 

.57343 

.34429 

.58928 

1.43476 

.60526 

1.53329 

.62135 

1.64097 

45 

46 

.57369 

.34573 

.58955 

1.43633 

.60552 

J.  53500 

.62162 

1.64285 

46 

47 

.57396 

.34718  i 

.58981 

1.43790 

.60579 

1.53672 

.62189 

1.64473 

47 

48 

.57422 

.34863  i 

.59008 

1.43948 

.60606 

1.53845 

.62216 

1.64662 

48 

19 

.57  IIS 

.35009 

.59034 

1.44106 

.60633 

1.54017 

.62243 

1.64851 

49 

:,i) 

.57475 

.35154 

.59061 

1.4-i264 

.60659 

1.54190 

.62270 

1.65040 

50 

51 

.57501 

.35300 

.59087 

1.44423 

.60686 

1.54363 

.62297 

1.65229 

51 

52 

.57527 

.35446 

.59114 

1.44582 

.60713 

1.54536 

.62324 

1.65419 

52 

58 

.5  75.")  4 

.35592 

.59140 

1.44741 

.60740 

1.54709 

.62351 

1.65609 

53 

:»i 

.57580 

.35738  | 

.59167 

1.44900 

.60766 

1.54883 

.62378 

1.65799 

54 

55 

.57606 

.35885  i 

.59194 

1.45059 

.60793 

1.55057 

.62405 

1.65989 

55 

56 

.57633 

.36031 

.59220 

1.45219 

.60820 

1.55231 

.62431 

1.66180 

5(5 

57 

.57659 

.36178 

.59247 

1.45378 

.60847 

1.55405 

.62458 

1.66371 

57 

58 

.57685 

1.36325 

.59273 

1.45539 

.60873 

1.55580 

.62485 

1.66563 

58 

59 

.57712 

1.36473 

.59300 

1.45699 

.60900 

1.55755 

.62512 

1.66755 

59 

GO 

.57738 

1.36620 

.59326 

1.45859 

.60927 

1.55930 

.62539 

1.66947 

60 

337 


TABLE  XIII.— VERSIXES  AND  EXSECAXTS. 


' 

68*                       69° 

70°                       71° 

| 
1 

Vers, 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers.      Exsec. 

0 

.62539  !  1.60947 

.64163 

1.79043 

.05798 

1.98880 

.67443  i  2.07155 

0 

1 

.62566 

1.67139 

.64190 

1.78864. 

.65825 

1.92014 

.07471 

2.07415 

1 

2 

.62593 

1.67332 

.(54218 

1.79466 

.65853 

1.92849 

.67498 

2.07-675 

2 

3 

.62620 

1.67525 

.64245 

1.79079 

.65880 

1.98083 

.07520 

2.07-936 

4 

.62647 

1.67718 

.04272 

1.79891 

.65907 

1.93318 

.67553 

2.08197 

4 

5 

.62674 

1.67911     I    .0429!) 

1.80104 

.65935 

1.93554 

.(>7:  .61   I  2.08459 

5 

6 

.62701 

1.68105 

.64326 

1.80318 

.65902 

1.93790 

.(;;v,08  i  2.08721 

( 

7 

.62728      1.6*299 

.64353 

1.80531 

.05989 

1.94026 

.67036 

2.08983 

'{ 

8 

.62755      1.68194 

.64381 

1.80746 

.66017 

1.94203 

.67063 

2.09246 

8 

9 

.62782      l.OSOS;) 

.64408 

1.80960 

.66044 

1.94500 

.67091      2.09510 

9 

10 

.62809 

1.68884 

.64435 

1.81175 

.60071 

1.94737 

.67718 

2.09774 

10 

11 

.62836 

1.69079 

.64462 

1.81390 

!   .66099 

1.94975 

.67746 

2.10038 

11 

12 

.62863 

1.69275 

.64489 

1.81605 

.66126 

1.95213 

.07773 

2.10303 

12 

13 

.62890 

1.69471 

.64517 

1.81821 

.66154 

1.95452 

.07801 

2.10508 

13 

14 

.62917 

1.69667 

.64544 

1.82037 

.66181 

1.95691 

.07829 

2.10834 

14 

15 

.62944 

1.69864 

.64571 

1.82254 

:    .66208 

1.95931 

.67856 

2.11101 

15 

16 

.62971 

1.70061 

.64598 

1.82471 

;   .66236 

1.90171 

.67884 

2.11367 

16 

17 

.62998 

1.70258 

.64625 

1.82688 

.66263 

1.96411 

.67911 

2.11635 

17 

18 

.63025 

1.70455  ! 

.64653 

1.82906 

.60290 

1.96652 

.67939 

2.11903 

18 

19 

.63052 

1.70653 

.64680 

1.88184 

.66318 

1.96893 

.U7966 

2.12171 

19 

20 

.63079 

1.70851 

.64707 

1.83342 

.66345 

1.97135 

.67994 

2.12440 

20 

21 

.63106 

1.71050 

.64734 

1.83561 

.66373 

1.97377 

.68021 

2.12709 

21 

22 

.631:33 

1.71249 

.64761 

1.88780 

.66400 

1.97619 

.08049 

2.12979 

22 

23 

.63161 

1.71448 

.64789 

1.83999 

.60427 

1.97862 

.68077 

2.13249 

23 

24 

.63188 

1.71647 

.64816 

1.84219 

.66455 

1.98100        .OHIO! 

2.13520 

24 

25 

.63215 

1.71847 

.64843 

1.84439        .06482 

1.98349 

.08132 

2.13791 

25 

26 

.63242 

1.72047 

.64870 

1.84659 

i    .66510 

1.98594 

!   .68159 

2.14063 

26 

27 

.63269 

1.72247 

.64898 

1.84880 

.66537 

1.98838 

.68187 

2.14335 

27 

28 

.63296 

1.72448 

.64925 

1.85102 

.06504 

1.99083 

.68214 

2.14608 

28 

29 

.63323 

1.72649 

.64952 

1.85323 

i    .66592 

1.99329 

.68242 

2.14881 

29 

30 

.63350 

1.72850 

.64979 

1.85545 

.66619 

1.99574 

.68270 

2.15155 

30 

31 

.63377 

1.73052 

.65007 

1.85767 

.66647 

1.99821 

.68297 

2.15429 

31 

32 

.63404 

1.73254 

.65034 

1.85990 

j   .66074 

2.00067 

.68325 

2.15704 

32 

33 

.63431 

1.73456 

.65061 

1.86213 

.66702 

2.00315 

.08352 

2.15979 

33 

34 

.63458 

1.73659 

.65088 

1.86437 

.66729 

2.00562 

.68380 

2.16255 

34 

35 

.63485 

1.73862 

.65116 

1.80C61 

.66756 

2.00810 

.68408     2.16531 

35 

36 

.63512 

1.74065 

.65143 

1.86885 

i    .66784 

2.01059 

.68435 

2.16808 

36 

37 

.63539 

1.74269 

165170 

1.87109 

.66811 

2.01308 

.68463 

2.17085 

37' 

38 

.63566 

1.74473  j 

.65197 

1.87884 

;   .66839 

2.01557 

.68490 

2.17363 

38 

39 

.63594 

1.74677 

.65225 

1.87560 

!   .66800 

2.01807 

.08518  !  2.17641 

39 

40 

.63621 

1.74881 

.65252 

1.87785 

.66894 

2.02057 

.68546 

2.17920 

40 

41 

.63648 

1.75086 

.65279 

1.88011 

.66921 

2.02308 

.68573 

2.18199 

41 

42 

.63675 

1.75292 

.65:306 

1.88388 

.66949 

2.02559 

.68601      2.18479 

42 

43 

.63702 

1.75497  i 

.65334 

1.88465 

.66976 

2.02810 

.68028 

2.18759 

43 

44 

.63729 

1.75703 

.65361 

1.88692 

.67003 

2.03062 

.68056 

2.19040 

44 

45 

.63756 

1.75909 

.65388 

1.88920 

.67031 

2.03315 

.68684 

2.19322 

45 

16 

.63783 

1.76116  i 

.65416 

1.89148 

.67058 

2.03568 

.68711      2.19604 

40 

17 

.63810 

1.7IJ323 

.65143 

1.89376 

.67-066 

2.03821 

.687'39     2.19886 

47 

48 

.63838 

1.76530  I 

.65470 

1.89605 

.07113 

2.04075 

.68707     2.20169 

48 

49 

.63865 

1.76737 

.65497 

1.89834 

.67141 

2.04329 

.68794 

2.20453 

49 

50 

.63892 

1.76945 

.65525 

1.90063 

.67168 

2.04584 

.68822 

2.20737 

50 

51 

.63919 

1.77154 

.65552 

1.90293 

.67196 

2.04839 

.68849 

2.21021 

51 

52 

.63946 

1.77:302 

.65579 

1.90524 

.67223 

2.05094 

.68877 

2.21306 

52 

53 

.63973  !  1.77571 

.65607 

1.90754 

.67351 

2.05350 

.68905 

2.21592 

53 

54 

.64000 

1  77780 

.65684 

1.90986 

67278 

2.05607 

.68932 

2.21878 

54 

55 

.64027 

1.77990 

.65661 

1.91217       .67:306 

2.  058(5  1    ,    .68900 

2.22165 

55 

56 

.64055 

1.78200 

.65689 

1.91449 

.67333  :  2.00121 

.68988 

2.22452 

56 

57 

64082 

1.78410 

.65716 

1.91681 

.67301      y.(MW7!) 

.69015 

2.22740 

57 

"58 

.64109 

1.78681 

.65743 

1.91914 

.(i7:is8    y.own? 

.69043 

2.2:3028  158 

59 

.64136 

1.7H832 

.65771 

1.JWN7      .<57H(»    2.or,s<)0 

.69071 

2.2:3317    59 

60 

.64163 

1.79043  i 

.65798      1.92380  I|    .67443     2.07155 

.69098 

2.23607  160 

TABLE  XIII.— VERSINES  AND  EXSECANTS. 


7 

2° 

7 

I 
3° 

7 

4° 

7 

5° 

, 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.69098 

2.23607 

.70763 

2.42030 

.72436 

2.62796 

.74118 

2.86370 

0 

1 

.09  120 

2.23897 

.70791 

2.42350 

.72404 

2.63164 

.74146 

2.86790 

1 

a 

.09154 

2.24187 

.70818 

2.42683 

.7'2492 

2  63533 

.74174 

2.87211 

g 

a 

.69181 

2.2447'8 

.70846 

2.43010 

.72520 

2.63903 

.74202 

2.87633 

3 

4 

.69209 

2.24770 

.70874 

2.43337 

.72548 

2.64274 

.74231 

2.88056 

4 

5 

.69:237 

2.25062 

.70902 

2.43666 

.72576 

2.64645 

.71259 

2.88479 

5 

G 

69264 

2.25355 

.70930 

2.43995 

.72604 

2.65018 

.7'4287 

2.88904 

6 

7 

.69292 

2.25648 

.70958 

2.44324 

.72632 

2.65391 

.74315 

2.89330 

7 

8 

.69320 

2.25942 

.70985 

2.44655 

.72660 

2.65765 

.74343 

2.89756 

8 

(J 

.69347 

2  26237 

.71013 

2.44986 

.72688 

2.66140 

.74371 

2.90184 

9 

10 

.69375 

2.26531 

.71041 

2.45317 

.72716 

2.60515 

.74399 

2.90613 

10 

11 

.69403 

2.26827 

.71069 

2.45650 

.72744 

2.66892 

.74427 

2.91042 

11 

12 

.09430 

2.27123 

.71097 

2.45983 

.72772 

2.67209 

.74455 

8.91473 

12 

18 

.69458 

2.27420 

.71125 

2.46316 

.72800 

2.07'647 

.74484 

2.91904 

13 

14 

.69486 

2.27717 

.71153 

2.46651 

.72828 

2.08025 

.74512 

2.92337 

14 

15 

.69514 

2.28015 

.71180 

2.46986 

.72856 

2.68405 

.74540 

2.92770 

15 

16 

.69541 

2.28313 

.71208 

2.47321 

.72884 

2.68785 

.74568 

2.93204 

1(1 

17 

.69569 

2.28612 

.71236 

2.47658 

.72912 

2.69167 

.74596 

2.9364C 

17 

IS 

.69597 

2.28912 

.71204 

2.47995 

.72940 

2.69549  . 

.74624 

2.94076 

18 

19 

.69624 

2.29212 

.71293 

2.48333 

.72968 

2.09931  1 

.74652 

2.94514 

1!) 

90 

.69652 

2.29512 

.71320 

2.48671 

.72996 

2.70315 

.74680 

2.94952 

80 

21 

.69680 

2.29814 

.71348 

2.49010 

.73024 

2.70700 

.74709 

2.95392 

81 

22 

.69708 

2.30115 

,71375 

2.49350 

.73052 

2.71085 

.74737 

2.95832 

22 

23 

.69735 

2.30418 

-71403 

2.49691 

.73080 

2.71471  j 

.74765 

2.'.)(i274 

23 

534 

.69763 

2.30721 

.71431 

2.50032 

.73108 

2.71858  ' 

.74793 

2.96716 

84 

85 

.69791 

2.31024 

.71459 

2.50374 

.73136 

2.72246 

.74821 

2.97160 

25 

86 

.69818 

2.31328 

.71487 

2.50716 

.73164 

2.72635 

.74849 

2.97604 

20 

27 

.69846 

2.31633 

.71515 

2.51060 

.73192 

2.73024 

.74878 

2.98050 

27 

28 

.69874 

2.31939 

.71543 

2.51404 

.73220 

2.73414 

.74906 

2.98497 

2S 

29 

.69902 

2.32244 

.71571 

2.51748 

.73248 

2.73806 

.74934 

2.98944 

29 

80 

.69929 

2.32551 

.71598 

2.52094 

.73276 

2.74198 

.74962 

2.99393 

80 

31 

.69957 

2.32858 

.71026 

2.52440 

,73304 

2.74591 

.74990 

2.99843 

:J1 

32 

.69985 

2.33166 

.71654 

2.52787 

.73332 

2.74984 

.75018 

3.00293 

82 

3:3 

.70013 

2.33474 

.71682 

2.53134 

.73360 

2.75379 

.75047 

3.00745 

83 

84 

.70040 

2.33783 

.71710 

2.53482 

.73388 

2.75775 

.75075 

3.01198 

31 

85 

.70068 

2.34092 

.71738 

2.53831 

.73416 

2.76171 

.75103 

3.01652 

85 

30 

.70096 

2.34403 

.71766 

2.54181 

.73444 

2.76568  : 

.75131 

3.02107 

36 

87 

.70124 

2.34713 

.71794 

2.54531 

.73472 

2.76966 

.75159 

3.02563 

87 

88 

.70151 

2.35025 

.71822 

2.54883 

.73500 

2.77305 

.75187 

3.03020 

88 

89 

.70179 

2.35336 

.71850 

2.55235 

.73529 

2.77765 

.75216 

3.03479 

89 

40 

.70207 

2.35649 

.71877 

2.55587 

.73557 

2.78166 

.75244 

3.03938 

4€ 

41 

.70235 

2.35962 

.71905 

2.55940 

.73585 

2.78568 

.75272 

3.0-1398 

41 

45 

.70263 

2.36270 

.71933 

2.56294 

.73613 

2.78970 

.75300 

3.04860 

42 

43 

.70290 

2.30590 

.71961 

2.56649 

.73641 

2.79374 

.75328 

3.05322 

43 

44 

.70318 

2.30905 

.71989 

2.57005 

.73009 

2.79778 

.75356 

8.05786 

41 

45 

.70:546 

2.37221 

.72017 

2.57361 

.73697 

2.80183 

.75385 

3.06251 

45 

46 

.70374 

2.37537 

.72045 

2.57718 

.73725 

^2.80589 

.75413 

3.06717 

40 

47 

.70401 

2.37854 

.7207'3 

2.5807'6 

.73753 

2.80996 

.75441 

3.07184 

47' 

4S 

.70429 

2.38171 

.72101 

2.58434 

.73781 

2.81404 

.75469 

3.07652 

4S 

49 

.70457 

2.38489 

.72129 

2.58794 

.73809 

2.81813 

.75497 

3.08121 

49 

50 

.70485 

2.38808 

.72157 

2.59154 

.73837 

2.82223 

.75526 

3.08591 

CO 

51 

.70513 

2.39128 

.72185 

2.59514 

.73865 

2.  82633 

.75554 

3.09063 

51 

52 

.70540 

2.39448 

.72213 

2.59876 

.73893 

2.8:3045 

.75582 

3.095&5 

52 

r,.-i 

.70568 

2.39768 

.72241 

2.60238 

.73921 

2.83457 

.75610 

3.10009 

53 

54 

.70596 

2.40089 

.72269 

2.00601 

.73950 

2.83871 

.75639 

3.10484 

51 

5.") 

.70624 

2.40411 

.72296 

2.60965 

.73978 

2.84285 

.75667 

8.10960 

55 

56 

.70652 

2.40734 

.72324 

2.61330 

.74006 

2.84700 

.75695 

3.11437 

50 

57 

.70679 

2.41057 

.72352 

2.61695 

.74034 

2.85116 

,75723 

3.11915 

57 

58 

.70707 

2.41381 

.72380 

2  62061 

.74062 

2.85533 

.75751 

3.12394 

58 

59 

.70735 

2.41705 

.72-408 

2.62428 

.74090 

2.85951 

.75780 

3.12875 

59 

GO 

.70763 

2.42030 

.72436 

2.62796  1 

.74118 

2.86370 

.75808 

3.13357 

80 

339 


TABLE  XIII.-VERSINES  AND  EXSECANTS. 


7 

6° 

7 

7° 

7 

8° 

7 

Qo 

Vers. 

Exsec. 

Vers. 

Exsec. 

Yers. 

Exsec. 

Yers. 

Exsec. 

0 

.75808 

3.13357 

.  77505 

3.44541 

i  .79209 

3.80973 

.80919 

4.24084 

0 

1 

.75836 

3.13839 

.77533 

3.45102 

.  .79237 

3.81033 

.80948 

4.24870 

1 

2 

.75864 

3.14323 

.77562 

!  3.45664 

.79200 

3.82294 

.80976 

4.2505S 

2 

3 

.75892 

3.14809 

.77590 

i  3.46228 

.79294 

3.82956 

.81005 

4.26448 

3 

4 

.75921 

3.15295 

.77618 

:  3.46793 

.79323 

3.83621 

.81033 

4.27241 

4 

5 

.75949 

3.15782 

.77647 

3.47360 

.79351 

3.84288 

.81062 

4.28036 

5 

6 

.75977 

3.16271 

.77675 

3.47938 

.79380 

3.84956 

.81090 

4.28833 

6 

7 

.76005 

3.16761 

.77703 

3.48498 

.79408 

3.85627 

.81119 

4.29634 

7 

8 

.76034 

3.17252 

.77732 

!  3.49069 

.79437 

3.86299 

.81148 

4.30436 

8 

g 

.76062 

3.17744 

.77760 

3.49642 

.79-165 

3.86973 

.81176 

4.31241 

9 

10 

.76090 

3.18238 

.77788 

3.50216 

.79493 

3.87649 

.81205 

4.32049 

10 

11 

.76118 

3.1S733 

.77817 

3.50791 

.79522 

3.88327 

.81233 

4.32859 

11 

13 

.76147 

3.19228 

.77845 

3.51368 

.79550 

3.89007 

.81202 

4.33671 

12 

13 

.76175 

3.19725 

.77874 

3.51947 

.79579 

3.89689 

i  .81290 

4.34486 

13 

14 

.76203 

3.20224 

.77902 

3.52527 

.79607 

3.90373 

.81319 

4.35304 

14 

15 

.76231 

3.20723 

.77930 

3.53109 

.79636 

3.91058 

.81348 

4.36124 

15 

18 

.76260 

3.21224 

.77959 

3.53692 

.79664 

3.91746 

:  .81376 

4.365)47 

16 

17 

.76288 

3.21726 

.77987 

3.54277 

.79093 

3.92436 

.81405 

4.37772 

17 

18 

.76316 

3.22229 

.78015 

3.54863 

.79721 

3.93128 

.81433 

4.38000 

18 

19 

.76344 

3.22734 

.78044 

3.55451 

.79750 

3.93821 

1  .81462 

4.3943') 

19 

20 

.76373 

3.23239 

.78072 

3.56041 

.79778 

3.94517 

.81491 

4.40263 

20 

21 

.76401 

3.23746 

.78101 

3.56632 

.79807 

3.95215 

.81519 

4.41099 

21 

2-2 

.76429 

3.24255 

.78129 

3.57224 

.79835 

3.95914 

.81548 

4.41937 

22 

88 

.76458 

3.24764 

.78157 

3.57819 

.79864 

3.96616 

'  .81576 

4.42778 

23 

24 

.76486 

3.25275 

!  .78186 

3.58414 

.79892 

3.97320 

.81605 

4.43022 

24 

X 

.76514 

3.25787 

.78214 

3.59012 

.79921 

3.98025 

1  .81633 

4.44468 

25 

96 

.76542 

3.26300 

.78242 

3.59611 

.79949 

3.98733 

.81662 

4.45317 

26 

2; 

.76571 

3.26814 

i  .78271 

3.60211 

.79978 

8.99443 

.81691 

4.46169 

27 

88 

.76599 

3.27330 

!  .78299 

3.60813 

.80006 

4.00155 

i  .81719 

4.47023 

28 

89 

.76627 

3.27847 

i  .78328 

3.61417 

.80035 

4.00869 

.81748 

4.47881 

29 

80 

.76655 

3.28366 

.7'8356 

3.62023 

.80063 

4.01585 

.81776 

4.48740 

SO 

81 

.76684 

3.28885 

.78384 

3.62630 

.80092 

4.02303 

.81805 

4.49603 

31 

82 

.76712 

3.29406 

.78413 

3.63238 

.80120 

4.03024 

.81834 

4.50468 

32 

88 

.76740 

3.29929 

.78441 

3.63849 

.80149 

4.03746 

<  .81862 

4.51337 

33 

84 

.76769 

3.30452 

1  .7847'0 

3.64461 

.80177 

4.04471 

:  .81891 

4.52208 

34 

86 

.76797 

3.30977 

.78498 

3.65074 

.80206 

4.05197 

.81919 

4.53081 

35 

30 

.76825 

3.31503 

|  .78526 

3.65690 

.80234 

4.05926 

i  .81948 

4.53958 

30 

87 

.76854 

3.32031 

.78555 

3.66307 

.80263 

4.06657 

.81977 

4.54837 

37 

88 

.76882 

3.32560 

.78583 

3.66925 

.80291 

4.07390 

.82005 

4.55720 

38 

39 

.76910 

3.33090 

.78612 

3.67545 

.80320 

4.08125 

.82034 

4.56605 

39 

40 

.76938 

3.33622 

.78640 

3.68167 

.80348 

4.08863 

.82063 

4.57493 

40 

41 

.76967 

3.34154 

.78669 

3.68791 

.80377 

4.09602 

.82091 

4.58383 

41 

42 

.76995 

3.34689 

.78697 

3.69417 

.80405 

4.10344 

.82120 

4.59277 

42 

43 

.77023 

3.35224 

.78725 

3.70044 

.80434 

4.11088 

.82148 

4.00174 

43 

44 

.77052 

3.35761 

.78754 

3.70673 

.80462 

4.11835 

.82177 

4.01073 

44 

45 

.77080 

3.36299 

.78782 

3.71303 

.80491 

4.12583 

.82206 

4.61U76 

45 

46 

.77108 

3.36839 

.78811 

3.71935 

.80520 

4.13334 

.82234 

4.62881 

46 

47 

.77137 

3.37380 

.78839 

3.72569 

.80548 

4.14087 

.82203 

4.03790 

47 

48 

.77165 

3.37923 

.78868 

3.73205 

.80577 

4.14842 

.82292 

4.047'01 

48 

49 

.77193 

3.38466 

.78896 

8.7'3843 

.80605 

4.15599 

.82320 

4.65616 

49 

60 

.77222 

3.39012 

.78924 

3.74482 

.80634 

4.16359 

.82349 

4.06533 

50 

51 

.77250 

3.39558 

.78953 

3.75123 

.80662 

4.17121 

.82377 

4.07451 

51 

52 

.77278 

3.40106 

>8981 

8.75766 

.80691 

4.17886 

.82406 

4.08377 

52 

58 

.77307 

3.40656 

.79010 

3.76411 

.80719 

4.18652 

.82435 

4.69304 

53 

64 

.77335 

3.41206 

.79038 

3.77057 

.80748 

4.19421 

.82463 

4.70234 

54 

55 

.77363 

3.41759 

.79067 

3.77705 

.8077'6 

4.20193 

.82492 

4.71166 

55 

66 

.77392 

3.42312 

.79095 

8.78365 

.80805 

4.20966 

.82521  , 

4.72102 

56 

57 

.77420 

3.42867 

.79123 

3.79007 

.80833 

4.21742 

82549 

4.73041 

57 

n 

.77448 

3.43424 

.79158 

3.79001 

.80868 

4.22521 

.82578 

4.73983 

58 

69 

.77477 

3.43982 

.79180 

3.80316 

.80891 

4.23801 

.82607 

4.74929 

59 

GO 

.77505 

3.44541 

.79209 

3.80973  1 

.80919 

4.24084 

.82635 

4.75877 

60 

340 


TABLE  XIII.-VERSINES  AND  EXSECANTS. 


' 

80° 

81°                            82° 

83° 

' 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.82635 

4.75877 

.84357 

5.39245       .86083 

6.18530 

.87813 

7.20551 

0 

1 

.82664 

4.76829 

.84385 

5.40422 

.86112 

6.20020 

.87842 

7.22500 

1 

2 

.82693 

4.77784 

.84414 

5.41602 

i   .86140 

6.21517 

.87871 

7.24457 

2 

3;    .83721 

4.78742 

.84443 

5.42787 

.86169 

6.23019 

.87900 

7.26425 

3 

4   .82:50 

4.79703 

.84471 

5.43977 

1   .86198 

6.24529 

.87929 

7.28402 

4 

51   .82:78 

4.80667 

.84500 

5.45171 

.86227 

6.26044 

.87957 

7.36388 

5 

6|   .85307 

4.81635 

.84529 

5.46369 

.86256 

6.27566 

.87986 

7.32384 

6 

7    .83336 

4.82606 

.84558 

5.47572 

.86284 

6.29095 

.88015 

7.34390 

7' 

8    .83864 

4.83581 

.84536 

5.48779 

.86313 

6.30630 

.88044 

7.36405 

8 

9     .&IS93 

4.84558 

.84615 

5.43991 

.86342 

6.32171 

.88073 

7.38431 

9 

10 

.82923 

4.85539 

.84644 

5.51208 

.86371 

6.33719 

.88102 

7.40466 

10 

11 

.82950 

4.86534 

.84673 

5.52429 

.86400 

6.35274 

.88131 

7.42511 

11 

12  i    .83979 

4.87511 

.84701 

5.53655 

.86428 

6.36835 

.88160 

7.44566 

12 

13 

.83003 

4.88503 

.84730     5.54886 

.86457 

6.38403 

.88188 

7.46632 

13 

14 

.83036 

4.89497 

.84759 

5.56121 

.86486 

6.39978 

.88217 

7.48707 

14 

15 

.83065 

4.90495 

.84788 

5.57361 

.86515 

6.41560 

.88246 

7.50793 

15 

16 

.83094 

4.91496 

.84816 

5.58606 

.86544 

6.43148 

.88275 

7.52889  il6 

17 

.83123 

4.92501 

.84845 

5.59855 

.86573 

6.44743 

.88304 

7.54996  !l7 

18 

.83151 

4.93509 

.84874 

5.61110 

.86601 

6.46346 

.88333 

7.57113    18 

11) 

.83180 

4.91521 

.84903 

6.  62309 

.86630 

6.47933 

.88362 

7.59241   ;19 

90 

.83208 

4.95536 

.84931 

5.63633 

.86659 

6.49571 

.88391 

7.61379  i  20 

•-•I 

.83237 

4.96555 

.84960 

5.64902 

.86688 

6.51194 

.88420 

7.6.3528   21 

.1.) 

.83266 

4.97577 

.84989 

5.66176 

.86717 

6.53825 

.88448 

7.65688  :22 

as 

.83294 

4.98603 

.85018 

5.67454 

.867'46 

6.54462 

.88477 

7.67859  (23 

24 

.83323 

4.99633 

.85046 

5.68738 

i   .86774 

6.56107 

.88506 

7.70041  ;24 

25 

.83352 

5.00666 

.85075 

5.70027 

!   .86803 

6.57759 

.88535 

7.72234  i25 

26 

.83380 

5.01703 

.85104 

5.71321 

.86832 

6.59418 

.88564 

7.74438 

2(5 

27 

.83409 

5.03743 

.85133 

5.72620 

.86861 

6.61085 

.88593 

7.76653 

27 

28    .83438 

5.03787 

.85162 

5.73924 

.86890 

6.62759 

.88623 

7.78880 

•'s 

29!    .83-107 

5.04834 

.85190 

5.75233 

.86919 

6.64441 

.88651 

7.81118  129 

30 

.83495 

5.05886 

.85219 

5.76547 

.86947 

6.66130 

.88680 

7.83367 

80 

81 

.83534 

5.06941 

.85248 

5.77866 

.86976 

6.67826 

.88709 

7.85628 

81 

83 

.83553 

5.03000 

.85277 

5.79191 

.87005 

6.69530 

.887'37 

7.87901 

32 

3:5 

.83581 

5.09062 

.85305 

5.80521 

.87-034 

6.71242 

.88766 

7.90186 

83 

84 

.83610 

5.10129 

.85334 

5.81856 

.87063 

6.72962 

.88795 

7.92482    34 

:J5 

.83639 

5.11199 

.85363 

5.83196 

.87092 

6.74689 

.88824 

7.94791 

35 

yr, 

.83667 

5.12273 

.85392 

5.84542 

.87120 

6.76424 

.88853 

7.97111 

36 

87 

.83696 

5.13330 

.85420 

5.85893 

.87149 

6.78167 

.88882 

7.99444 

87 

3H 

.83735 

5.14433 

.85449 

5.87250 

.87178 

6.79918 

.88911 

8.01788 

38 

30 

.83754 

5.15517 

.85478 

5.88612 

.87207 

6.81677 

.88940 

8.04146 

89 

40 

.83782 

5.16607 

.85507 

5.89979 

.87236 

6.83443 

.88969 

8.06515 

40 

41 

.83811 

5.17700 

.85536 

5.91352 

.87265 

6.85218 

.88998 

8.08897 

41 

42  !   .83840 

5.18797 

.85564 

5.92731 

.87294 

6.87001 

.89027 

8.11292 

4-2 

43 

.83868 

5.19898 

.85593 

5.94115 

.87322 

6.88792 

.89055 

8.13699 

ta 

U 

.83897 

5.21004 

.85622 

5.95505 

.87351 

6.90592 

.89084 

8.16120 

44 

45 

.83936 

5.23113 

.85651 

5.96900 

.87380 

6.92400 

.89113 

8.18553 

45 

46     .83954 

5.23226 

.85680 

5.98301 

i   .87409 

•6.94216 

.89142 

8.20999 

46 

47  1   .830S3 

5.24343 

.85708 

5.997'08 

i   .87438 

6.96040 

.89171 

8.23459 

41' 

48  :   .84012 

5.25464 

.85737 

6.01120 

.87467 

6.97873 

.89200 

8.25931    48 

49  |   .81041 

5.26590 

.85766 

6.02538 

.87496 

6.99714 

.89229 

8.28417  :49 

50 

.84069 

5.27719 

.85795 

6.03962 

.87524 

7.01565 

.89258 

8.30917    50 

51 

.84098 

5.28853 

.85823 

6.05392 

.87553 

7.03423 

.89287 

8.33430    51 

52 

.84127 

5.29991 

:  85852 

6.06828 

.87582 

7.05291 

.89316 

8.35957 

52 

53 

.84155 

5.31133 

.85881 

6.08269 

.87611 

7.07167 

.89345 

8.38497 

53 

54 

.84184 

5.32279 

.85910 

6.09717 

.87640 

7.09052 

.89374 

8.41052 

51 

55 

.84213 

5.33429 

.85939 

6.11171 

.87669 

7.10946 

.89403 

8.43620 

55 

56 

.84242 

5.34584 

.85967 

6.12630 

.87698 

7.12849 

.89431 

8.46203 

56 

57 

.84270 

5.35743 

.85996 

6.14096 

.87726 

7.14760 

.89460 

8.48800 

57 

58 

.84299 

5.36906 

.86025 

6.15568 

.87755 

7.16681 

.89489 

8.51411 

58 

58 

.84328 

5.38073 

.86054 

6.17046 

.87784 

7.18612 

.89518 

8.54037 

59 

CO 

.84357 

5.39345 

.86083 

6.18530 

.87813 

7,20551 

.89547 

8.56677 

60 

341 


TABLE  xin.— VERSINES  AND  EXSECANTS. 


/ 

84° 

85° 

86° 

' 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.89547 

8.5(3677 

.91284 

10.47371 

.93024 

13.33559 

0 

1 

.89576 

8.59332 

.91313 

10.51199 

.93053 

13.39547 

1 

2 

.89605 

8.62002 

.91342 

10.55052 

.93082 

13.45586 

2 

3 

.89634 

8.64687 

.91371 

10.58932 

.93111 

13.51676 

3 

4 

.89663 

8.67387 

.91400 

10.62837 

.93140 

13.57817 

4 

5 

.89692 

8.70103 

.91429 

10.66769 

.93169 

13.64011 

5 

6 

.89721 

8.72833 

.91458 

10.7'0728 

.93198 

13.70258 

6 

7 

.89750 

8.75579 

.91487 

10.74714 

.93227 

13.76558 

7 

8 

.89779 

8.78341 

.91516 

10.787'27 

.93257 

13.82913 

8 

9 

.89808 

8.81119 

.91545 

10.82768 

'  .93286 

13.89823 

9 

10 

.89836 

8.83912 

.91574 

10.86837 

.93315 

13.957B8 

10 

11 

.89865 

8.86722 

.91603 

10.90934 

.93344 

14.02310 

11 

12 

.89894 

8.89547 

.91632 

10.95060 

.98373 

14.08890 

12 

13 

.89923 

8.92389 

.91661 

10.99214 

.93102 

14.15527 

13 

14 

.89952 

8.95248 

.91690 

11.03397 

.93431 

14.22223 

14 

15 

.89981 

8.98123 

.91719 

11.07610 

.93460 

14.28979 

15 

16 

.90010 

9.01015 

.91748 

11.11852 

.93489 

14.35795 

16 

17 

.90039 

9.03923 

.91777 

11.16125 

.93518 

14.42672 

17 

18 

.90068 

9.06849 

.91806 

11.20427 

.93547 

14.49611 

18 

19 

.90097 

9.09792 

.91835 

11.24761 

.93576 

14.56614 

19 

20 

.90126 

9.12752 

.91864 

11.29125 

.93605 

14.63679 

20 

21 

.90155 

9.15730 

.91893 

11.33521 

.93634 

14.70810 

21 

22 

.90184 

9.18725 

.91922 

11.37948 

.93663 

14.78005 

22 

23 

.90213 

9.21739 

.91951 

11.42408 

.93692 

14.85268 

23 

24 

.90242 

9.24770 

.91980 

11.40900 

.93721 

14.92597 

24 

25 

.90271 

9.27819 

.92009 

11.51424 

.93750 

14.99995 

25 

26 

.90300 

9.30887 

.92038 

11.55982 

.93779 

15.07462 

26 

27 

.90329 

9.33973 

.92067 

11.60572 

.93808 

15.14999 

27 

28 

.90358 

9.37077 

.92096 

11.65197 

.93837 

15.22607 

28 

29 

.90386 

9.40201 

.92125 

11.69856 

.93866 

15.30287 

29 

30 

.90415 

9.43343 

.92154 

11.74550 

.93895 

15.38041 

30 

31 

.90444 

9.46505 

.92183 

11.79278 

.93924 

15.45869 

31 

32 

.90473 

9.49685 

.92212 

11.84042 

.93953 

15.53772 

32 

33 

.90502 

9.52886 

.92241 

11.88841 

.93982 

15.61751 

33 

34 

.90531 

9.56106 

.92270 

11.93677 

.94011 

15.69808 

34 

35 

.90560 

9.59346 

.92299 

11.98549 

.94040 

15.77944 

35 

36 

.90589 

9.62605 

.92328 

12.03458 

.94069 

15.86159 

36 

37 

.90618 

9.65885 

.92357 

12.08040 

.94098 

15.94456 

37 

38 

.90647 

9.69186 

.92386 

12.13388 

.94127 

16.02835 

38 

39 

.90676 

9.72507 

.92415 

32.18411 

.94156 

16.11297 

39 

40 

.90705 

9.75849 

.92444 

12.23472 

.94186 

16.198-13 

40 

41 

.90734 

9.79212 

.92473 

12.28572  1 

.94215 

16.28476 

41 

42 

.90763 

9.82596 

.92502 

12.33712 

.94244 

16.37196 

4:3 

43 

.90792 

9.86001 

.92531 

12.38891 

.94273 

16.46005 

43 

44 

.90821 

9.89428 

.92560 

12.44112 

.94302 

16.54903 

41 

45 

.90850 

9.92877 

.92589 

12.49373 

.94331 

16.63893 

45 

46 

.90879 

9.96348 

.92618 

12.54676 

.94360 

16.72975 

46 

47 

.90908 

9.99841 

.92647 

12.60021 

.94389 

16.82152 

47 

48 

.90937 

10.03356 

.92676 

12.65408 

.91418 

16.91424 

48 

49 

.90966 

10.06894 

.92705 

12.70838 

.94447 

17.00794 

49 

50 

.90995 

10.10455 

.93734 

12.76312 

.94476 

17.10262 

50 

51 

.91024 

10.14039 

.92763 

12.81829 

.94505 

17.19830 

51 

52 

.91053 

10.17646 

.92792 

12.87391 

.94534 

17.29501 

52 

53 

.91082 

10.21277 

.92821 

12.92999 

.94563 

17.39274 

53 

54 

.91111 

10.24932 

.92850 

12.98651 

.94592 

17.49153 

64 

55 

.91140 

10.28610  i 

.92879 

13  04350 

.94621 

17.59139 

55 

56 

.91169 

10.32313 

.92908 

13.1001)0 

.94050 

17.69233 

56 

-  57 

.91197 

10.36040 

.92937 

13.15889 

.94679 

17.79438 

57 

58 

.91226 

10.39792 

.92966 

13.21730 

.94708 

17.89755 

58 

59 

.91255 

10.43569 

.92995 

13.27630 

.94737 

18.00185 

59 

60 

.91284 

10.47371 

.93024 

13.33559 

.94766 

18.10732 

60 

343 


TABLE  XHI.— VERSINES  AND  EXSECANTS. 


/ 

87° 

88° 

89° 

/ 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.94766 

18.10732 

.96510 

27.65371 

.98255 

56.29869 

0 

1 

.94795 

18.21397 

.96539 

27.89440 

.98284 

57.26976 

1 

2 

.94825 

18  32182 

.96568 

28.13917 

.98313 

58.27'431 

2 

3 

.94854 

18.43088 

.96597 

28.38812 

.98342 

59.31411 

3 

4 

.94883 

18.54119 

.96626 

28,64137 

.98371 

60.39105 

4 

5 

.91912 

18.65275 

.96655 

28.89903 

.98400 

61.50715 

5 

6 

.91941 

18.76560 

.96684 

29.16120 

.98429 

62.66460 

6 

7 

.94970 

18.87976 

.96714 

29.42802 

.98458 

63.86572 

7 

8 

.94999 

18.99524 

.96743 

29.69960 

.98487 

65.11304 

8 

9 

.95028 

19.11208 

.96772 

29.97607 

.98517 

66.40927 

9 

10 

.95057 

19.23028 

.96801 

30.25758 

.98546 

67.75736 

10 

11 

.95086 

19.34989 

.96830 

30.54425 

.98575 

69.16047 

11 

12 

.95115 

19.47'093 

.96859 

30.83(523 

.98604 

70.62285 

12 

13 

.95144 

19.59341 

.96888 

31.13366 

.98633 

72.14583 

13 

14 

.95173 

19.71737 

.96917 

31.43671 

.98662 

73.7'3586 

14 

15 

.95202 

19.84283 

.96946 

31.74554 

.98691 

75.39655 

15 

16 

.95231 

19.96982 

.96975 

32.06030 

.98720 

77.13274 

16 

17 

.95260 

20.09838 

.97004 

32.38118 

.98749 

78.94968 

17 

18 

.95289 

20.22852 

.97033 

32.70835 

.98778 

80.85315 

18 

19 

.95318 

20.36027 

.97062 

33.04199 

.98807 

82.84947 

19 

20 

.95347 

20.49368 

.97092 

33.38232 

.98836 

84.94561 

20 

21 

.95377 

3C.  62876 

.97121 

33.72952 

.98866 

87.14924 

21 

22 

.95406 

20.76555 

.97150 

34.08380 

.98895 

89.46886 

22 

23 

.95485 

20.90409 

.97179 

34.44539 

.98924 

91.91387 

23 

21 

.95464 

21.04440 

.97208 

34.81452 

.98953 

94.49471 

24 

95 

.95493 

21.18653 

.97237 

35.19141 

.98982 

97.22303 

25 

86 

.95522 

21.33050 

.97266 

35.57633 

.99011 

100.1119 

26 

27 

.95551 

21.47635 

.97295 

35.96953 

.99040 

103.1757 

27 

23 

.95580 

21.62413 

.97324 

36.37127 

.99069 

106.4311 

28 

29 

.95609 

21.77386 

.97353 

36.78185 

.99098 

109.8966 

29 

30 

.95638 

21.92559 

.97382 

37.20155 

.99127 

113.5930 

30 

31 

.95667 

22.07935 

.97411 

37.63068 

.99156 

117.5444 

81 

33 

.95696 

22.23520 

.97440 

38.06957 

.99186 

121.7780 

32 

33 

.95725 

22.39316 

.97470 

38.51855 

.99215 

126.3253 

33 

34 

.95754 

22.55329 

.97499 

38.97797 

.99244 

131.2223 

34 

35 

.95783 

22.71563 

.97528 

39.44820 

.99273 

136.5111 

35 

36 

.95812 

22.88022 

.97557 

39.92963 

.99302 

142.2406 

36 

37 

.95842 

23.04712 

.97'586 

40.42266 

.99331 

148.4684 

37 

38 

.95871 

23.21637 

.97615 

40.92772 

.99360 

155.2623 

38 

39 

.95900 

23.38802 

.97644 

41.44525 

.99389 

162.7033 

39 

40 

.95929 

23.56212 

.97673 

41.97571 

.99418 

170.8883 

40 

41 

.95958 

23.73873 

.97702 

42.51961 

.99447 

179.9350 

41 

42 

.95987 

23.91790 

.97731 

43.07746 

.99476 

189.9868 

42 

43  l  .96016 

24.09969 

.97760 

43.64980 

.99505 

201.2212 

43 

44   .96045 

24.28414 

.97789 

44.23720 

.99535 

213.8600 

44 

45  !  .96074 

24.47134 

.97819 

44.84026 

.99564 

228.1839 

45 

4(5    .96103 

24.66132 

.97848 

45.459GS 

.99593 

244.5540 

46 

47 

.96132 

24.85417 

.97877 

46.09596 

.99622 

263.4427 

47 

48 

.96161 

25.04994 

.97906 

46.74997 

.99651 

285.4795 

48 

49 

.96190 

25.24869 

.97935 

47.42241 

.99680 

311.5230 

49 

50 

.90219 

25.45051 

,07964 

48.11406 

.99709 

342.7752 

50 

51 

.96248 

25.65546 

.97993 

48.82576 

.99738 

380.9723 

51 

52 

.96277 

25.86360 

.98022 

49.55840 

.99767 

428.7187 

52 

53 

.96307 

26.07503 

.98051 

50.31290 

.99796 

490.1070 

53 

54 

.96336 

26.28981 

.98080 

51.09027 

.99825 

571.9581 

54 

55 

.96365 

26.50804 

.98109 

51.89156 

.99855 

686.5496 

55 

56 

.96394 

26.72978 

.98138 

52.71790 

.99884 

858.4369 

56 

57 

.96423 

26.95513 

.'.tSKiS 

53.57046 

.99913 

1144.916 

57 

58 

.96452 

27.18417 

.98197 

54.45053 

.99942 

1717.874 

58 

59 

.96481 

27.41700 

.98226 

55.35946 

.99971 

3436.747 

59 

60 

.96510 

27.65371 

.98255 

56.29869 

1.00000 

Infinite 

60 

343 


TABLE  XIV.- CUBIC  YARDS  PER  100  FEET.      SLOPES   '4  :  1. 


Depth 

Base 
12 

Base 
14 

Base 
16 

Base 
18 

Base 
22 

Base 
24 

Base 
26 

Base 
28 

1 

45 

53 

60 

68 

82  |   90 

97 

105 

2 

93 

107 

122 

137 

167 

181 

196 

211 

3 

142 

163 

186 

208 

253 

275 

297 

319 

4 

193 

222 

252 

281 

341 

370 

400 

430 

5 

245 

282 

319 

356 

431 

468 

505 

542 

6 

300 

344 

389 

433 

522 

567 

611 

656 

7 

356 

408 

460 

612 

616 

668 

719 

771 

8 

415 

474 

533 

593 

711 

770 

830 

889 

9 

475 

542 

608 

675 

808 

875 

942 

1008 

10 

537 

611 

685 

759 

907 

981 

1056 

1130 

11 

601 

682 

764 

845 

1008 

-  1090 

1171 

1253 

12 

667 

756 

844 

933 

1111 

1200 

1289 

1378 

13 

734 

831 

926 

1023 

1216 

1312 

1408 

1505 

14 

804 

907 

1010 

1115 

1322 

1426 

1530 

1633 

15 

875 

986 

1096 

1208 

1431 

1542 

1653 

1764 

16 

948 

1067 

1184 

1304 

1541 

1659 

1778 

1896 

17 

1023 

1149 

1274 

1401 

1653 

1779 

1905 

2031 

18 

1100 

1233 

1366 

1500 

1767 

1900 

2033 

2167 

19 

1179 

1319 

1460 

1601 

1882 

2023 

2164 

2305 

20 

1259 

1407 

1555 

1704 

2000 

2148 

2296 

2444 

21 

1342 

1497 

1653 

1808 

2119 

2275 

2431 

2586 

22 

1426 

1589 

1752 

1915 

2241 

2404 

2567 

2730 

23 

1512 

1682 

1853 

2023 

2364 

2534 

2705 

2875 

24 

1600 

1778 

1955 

2133 

2489 

2667 

2844 

3022 

25 

1690 

1875 

2060 

2245 

2616 

2801 

2986 

3171 

26 

1781 

1974 

2166 

2359 

2744 

2937 

3130 

33:22 

27 

1875 

2075 

2274 

2475 

2875 

3075 

3275 

3475 

28 

1970 

2178 

2384 

2593 

3007 

3215 

3422 

30:30 

29 

2068 

2282 

2496 

2712 

3142 

3358 

3571 

3786 

30 

2167 

2389 

2610 

2833 

3278 

3500 

3722 

3944 

31 

2268 

2497 

2726 

2956 

3416 

3645 

3875 

4105 

32 

2370 

2607 

2844 

3081 

3556 

3793 

4030 

4207 

33 

2475 

2719 

2964 

3208 

3697 

3942 

4186 

4431 

34 

2581 

2833 

3085 

3337 

3841 

4093 

4344 

4596 

35 

2690 

2949 

3208 

3468 

3986 

4245 

4505 

4764 

36 

2800 

3067 

3333 

3600 

4133 

4400 

4667 

4933 

37 

2912 

3186 

3460 

3734 

4282 

4556 

4831 

5105 

88 

3026 

3307 

3589 

3870 

4433 

4715 

4996 

527'8 

39 

3142 

3431 

3719 

4008 

4586 

4875 

5164 

5-153 

40 

3259 

3556 

3852 

4148 

4741 

5037 

5333 

5630 

41 

3379 

3682 

3986 

4290 

4897 

5201 

5505 

5808 

42 

3500 

3811 

4122 

4433 

5056 

5367 

5678 

5989 

43 

3623 

3942 

4260 

4579 

5216 

5534 

5853 

6171 

44 

3748 

4074 

4400 

4726 

5378 

5704 

6030 

6356 

45 

3875 

4208 

4541 

4875 

5542 

5875 

6208 

6542 

46 

4004 

4344 

4684 

5026 

5707 

6048 

6389 

6730 

47 

4134 

4482 

4830 

5179 

5875 

6223 

6571 

6919 

48 

4267 

4(522 

4978 

5333 

6044 

6400 

6756 

7111 

49 

4401 

4764 

5127 

5490 

6216    6579 

6942 

7305 

50 

4537 

4907 

5278 

5648 

6389 

6759 

7130 

7500 

51 

4675 

5053 

5430 

5808 

6564 

6942 

7319 

7697 

52 

4815 

5200 

5584 

597'0 

6741 

7126 

7511 

7896 

53 

4956 

5349 

5741 

6134 

6919 

7312 

7705 

8097 

54 

5100 

5500 

5900 

6300 

7100 

7500 

7900 

8300 

55 

5245 

5653 

6060 

6468 

7282 

7690 

8097 

&505 

56 

5393 

5807 

6222 

6637 

7467 

7881 

8296 

8711 

57 

5542 

5964 

6386 

6808 

7653 

8075 

8497 

8919 

58 

5693 

6122 

6552 

6981 

78-11 

8270 

8700 

9130 

59 

5845 

6282 

6719 

7156 

8031 

8468 

8905 

9342 

60 

6000 

6444 

6889 

7333 

8222 

8667 

9111 

9556 

344 


TABLE  xiv.  -CUBIC  YARDS  PER  100  FEET.    SLOPES 


D^pth 

Base 
12 

Base 
14 

Base 
16 

Base 
18 

Base 
22 

Base 
24 

Base 
26 

Base 
28 

1 

46 

54 

61 

69 

83 

91 

98 

106 

2 

96 

111 

126 

141 

170 

185 

200 

215 

3 

150 

172 

194 

217 

261 

283 

306 

328 

4 

207 

237 

267 

296 

356 

385 

415 

444 

5 

269 

306 

343 

380 

454 

491 

528 

565 

6 

333 

378 

422 

467 

556 

600 

644 

689 

7 

402 

454 

506 

557 

661 

713 

765 

817 

8 

474 

533 

593 

652 

770 

&30 

889 

948 

9 

550 

617 

683 

750 

883 

950 

1017 

1083 

10 

630 

704 

778 

852 

1000 

1074 

1148 

1222 

11 

713 

794 

876 

957 

1120 

1202 

1283 

1365 

12 

800 

889 

978 

1067 

1244 

1333 

1422 

.  1511 

13 

891 

987 

1083 

1180 

1372 

1469 

1565 

1661 

It 

985 

1089 

1193 

1296 

1504 

1607 

1711 

1815 

15 

1083 

1194 

1306 

1417 

1639 

1750 

1861 

1972 

16 

1185 

1304 

1422 

1541 

1779 

1896 

2015 

2133 

17 

1291 

1417 

1543 

1669 

1920 

2046 

2172 

2298 

18 

1400 

1533 

1667 

1800 

2067 

2200 

2333 

2467 

19 

1513 

1654 

1794 

1935 

2217 

2357 

2498 

2639 

20 

1630 

1778 

1926 

2074 

2370 

2519 

2667 

2815 

21 

1750 

1906 

2061 

2217 

2528 

2683 

2839 

2994 

22 

1874 

2037 

2200 

2363 

2689 

2852 

3015 

3178 

23 

2002 

2172 

2343 

2513 

2854 

3024 

3194 

3365 

24 

2133 

2311 

2489 

2667 

3022 

3200 

3378 

3556 

25 

2269 

2454 

2639 

2824 

3194 

3380 

3565 

3750 

26 

2407 

2600 

2793 

2985 

3370 

35C3 

3756 

3948 

27 

2550 

2750 

2950 

3150 

&550 

3750 

3950 

4151 

28 

2696 

2904 

3111 

3319 

3733 

3941 

4148 

4356 

29 

2846 

3061 

3276 

3491 

3920 

4135 

4350 

4565 

30 

3000 

3222 

3444 

3667 

4111 

4333 

4556 

4778 

31 

3157 

3387 

3617 

3846 

4306 

4535 

4765 

4994 

32 

3319 

3556 

3793 

4030 

4504 

4741 

4978 

5215 

33 

3483 

3728 

397'2 

4217 

4706  ' 

4950 

5194 

5439 

34 

3652 

3904 

4156 

4407 

4911 

5163 

5415 

5G67 

35 

3824 

4083 

4343 

4602 

5120 

5380 

5639 

5898 

36 

4000 

4267 

4533 

4800 

5333 

5600 

5867 

6133 

37 

4180 

4454 

4728 

5002 

5550 

5824 

6098 

6372 

38 

4363 

4644 

4926 

5207 

5770 

6052 

6333 

6615 

39 

4550 

4839 

5128 

5417 

5994 

6283 

6572 

6861 

40 

4741 

5037 

5333 

5630 

6222 

6519 

6815 

7111 

41 

4935 

5239 

5543 

5846 

6454 

6757 

7061 

7365 

40 

5133 

5444 

5756 

6067 

6689 

7000 

7311 

7622 

43 

5335 

5654 

5972 

6291 

6928 

7246 

7565 

7883 

44 

5541 

5867 

6193 

6519 

7170 

7496 

7822 

8148 

45 

5750 

6083 

6417 

6750 

7417 

7750 

8083 

8417 

46 

5963 

6304 

C644 

6985 

7667 

8007 

8348 

8689 

47 

6180 

6528 

6876 

7224 

7920 

8269 

8617 

8965 

48 
49 

6400 
6624 

6756 

6987 

7111 
7350 

m 

8178 
843M 

8889 
9165 

9244 
9528 

50 

6852 

7222 

7593 

7963 

87G4 

9074 

9444 

9815 

51 

7083 

7461 

7839 

8217 

8972 

9350 

9728 

10106 

52 

7319 

7704 

8089 

847'4 

9244 

9630 

10015 

10400 

53 

7557 

7950 

8343 

8735 

9520 

9913 

10306 

10698 

54 

7800 

8200 

8000 

9000 

9800 

10200 

10600 

11000 

55 

8046 

8454 

8861 

9269 

10083 

10491 

10898 

11306 

56 

8296 

8711 

9126 

9541 

10370 

10785 

11200 

11615 

57 

8550 

8972 

9394 

9817  ' 

10661 

11083   11506 

11928 

58 

8807 

9237 

9667 

10096 

10956 

11385  i  11815 

12244 

59 

9069 

9506 

9943 

10380 

11254 

11691   12128 

12565 

60 

9333 

9778 

10222 

10667 

11556 

12000 

12444 

12889 

345 


TABLE  XIV. -CUBIC  YARDS  PER  100  FEET.      SLOPES  1  •  1. 


Depth 

Base 
12 

Base 
14 

Base 
16 

Base 
1.8 

Base 
20 

Base 
28 

Base- 
30 

Base 
32 

1 

48 

56 

03 

70 

78 

107 

115 

122 

2 

104 

119 

138 

148 

163 

222 

237 

252 

3 

167 

189 

211 

283 

256 

344 

367 

389 

4 

237 

267 

296 

326 

356 

474 

504 

5133 

5 

315 

352 

389 

426  • 

463 

611 

648 

685 

6 

400 

444 

489 

533 

578 

756 

800 

844 

7 

493 

544 

596 

648 

700 

907 

959 

1011 

8 

593 

652 

711 

770 

830 

1067 

1126 

1185 

9 

700 

767 

833 

900 

967 

1233 

1300 

1367 

10 

815 

889 

963 

1037 

1111 

1407 

1481 

1556 

11 

937 

1019 

1100 

1181 

1263 

1589 

1670 

1752 

12 

1067 

1156 

1244 

13:33 

1422 

1778 

1867 

1956 

13 

1204 

1300 

1396 

1493 

1589 

1974 

2070 

2167 

14 

1348 

1452 

1556 

1659 

1763 

2178 

2281 

2385 

15 

1500 

1611 

1722 

1833 

1944 

2389 

2500 

2611 

16 

1659 

1778 

1896 

2015 

2133 

2607 

2726 

2844 

17 

1826 

1952 

2078 

2204 

2330 

2833 

2959 

3085 

18 

2000 

2133 

2267 

2400 

2533 

3067 

3200 

3333 

19 

2181 

2322 

2463 

2604 

2744 

3307 

3448 

3589 

20 

2370 

2519 

2667 

2815 

2963 

3556 

3704 

3852 

21 

2567 

2722 

2878 

3033 

31  R9 

3811 

3967 

4122 

22 

2770 

2933 

3096 

3259 

34-^2 

4074 

4237 

4444 

23 

2981 

3152 

3322 

3493 

3663 

4344 

4515 

4685 

24 

3200 

3378 

3556 

8733 

3911 

4622 

4800 

4978 

25 

3426 

3611 

3796 

3981 

4167 

4907 

5093 

5278 

26 

3659 

3852 

4044 

4237 

4430 

5200 

5393 

5585 

27 

3900 

4100 

4300 

4500 

4700 

5500 

5700 

5900 

28 

4148 

4356 

4563 

4770 

4978 

5807 

6015 

6222 

29 

4404 

4619 

4833 

5048 

5263 

6122 

6337 

6552 

30 

4667 

4889 

5111 

5333 

5556 

6444 

6667 

6889 

31 

4937 

5167 

5396 

5626 

5856 

6774 

7004 

9X83 

32 

5215 

5452 

5689 

5926 

6163 

7111 

7318 

7585 

33 

5500 

5744 

5989 

6233 

6478 

7456 

1700 

7944 

34 

5793 

6044 

6296 

6518 

6800 

7807 

8059 

8311 

35 

6093 

6352 

6611 

6870 

7130 

8167 

8426 

8685 

36 

6400 

6GG7 

69:33 

7200 

7467 

8533 

8800 

90G7 

37 

6715 

6989 

7263 

7537 

7811 

8907 

9181 

9456 

38 

7037 

7319 

7600 

7881 

8163 

9289 

9570 

9852 

39 

7367 

7656 

7944 

8233 

8522 

9678 

9967 

10256 

40 

7704 

8000 

8296 

8593 

8889 

10074 

10370 

10667 

41 

8048 

8352 

8656 

8959 

9263 

10478 

10781 

11085 

42 

8400 

8711 

9022 

9333 

9644 

10889 

11300 

11511 

43 

8759 

9078 

9396 

9715 

10033 

11307 

11626 

11944 

44 

9126 

9452 

9778 

10104 

10430 

11733 

12059 

12385 

45 

9500 

9833 

10167 

10500 

10833 

12167 

12500 

12833 

46 

9881 

10222 

10563 

10904 

11244 

12607 

12948 

13289 

47 

10270 

10619 

10967 

11315 

11663 

13C56 

1:3404 

13752 

48 

10667 

11022 

11378 

11733 

12089 

13511 

13867 

14223 

49 

11070 

11433 

11796 

12159 

13523 

13974 

14337 

14700 

50 

11481 

11852 

12222 

12593 

12963 

14444 

14815 

15185 

51 

11900 

12278 

12656 

13033 

13411 

14922 

15300 

15678 

52 

12326 

12711 

13096 

13481 

13867 

15407 

15793 

16178 

53 

12759 

13152 

13544 

13937 

14330 

15900 

16293 

16685 

54 

13200 

13600 

14000 

14400 

14800 

16400 

16800 

17200 

55 

13648 

1405(1 

14463 

14870 

15278 

16907 

17315 

17722 

56 

14104 

14519 

14933 

15348 

15763 

17423 

17837 

1R252 

57 

14567 

14989 

15411 

15833 

16256 

17944 

18367 

18789 

58 

15037 

15467 

15896 

16326 

16756 

18474 

18904 

19333 

59 

15515 

15952 

16389 

KiSei) 

17263 

19011 

10118 

19885 

60 

16000 

16444 

16889 

17333 

17778 

19556 

20000 

20444 

346 


TABLE  XIV.  —CUBIC  YARDS  PER  100  FEET.      SLOPES 


:  1. 


Depth 

Base 
12 

Base 
14 

Base 
16 

Base 
18 

Base 
20 

Base 
28 

Base 
30 

Base 
32 

1 

50 

57 

65 

72 

80 

109 

117 

124 

2 

111 

126 

141 

156 

170 

230 

244 

259 

3 

183 

206 

228 

250 

272 

361 

383 

406 

4 

267 

296 

326 

356 

385 

504 

533 

563 

5 

301 

398 

435 

472 

509 

657 

694 

731 

6 

467 

511 

556 

600 

644 

822 

867 

911 

7 

583 

635 

687 

739 

791 

998 

1050 

1102 

8 

711 

770 

830 

889 

948 

1185 

1244 

1304 

9 

850 

917 

983 

1050 

1116 

1883 

1450 

1517 

10  ' 

1000 

1074 

1148 

1222 

1296 

1593 

1667 

1741 

11 

1161 

1243 

1324 

1406 

1487 

1813 

1894 

1976 

12 

1888 

1422 

1511 

1600 

1689 

2044 

2133 

2222 

13 

1517 

1613 

1709 

1806 

1902 

2287 

2383 

'  2480 

14 

1711 

1815 

1919 

2022 

2126 

2541 

2644 

2748 

15 

1917 

2028 

2139 

2250 

2361 

2806 

2917 

3028 

16 

2133 

2252 

2370 

2489 

2607 

3081 

3200 

3319 

17 

2361 

2487 

2613 

2739 

2865 

3369 

3494 

3620 

18 

2600 

2733 

2867 

3000 

3133 

3(567 

3800 

3933 

19 

2850 

2991 

3131 

3272 

3413 

3976 

4117 

4257 

20 

3111 

3259 

3407 

3556 

3704 

4296 

4444 

4592 

21 

3383 

3539 

3694 

3850 

4005 

4628 

4783 

4939 

22 

3667 

3830 

3993 

4156 

4318 

4970 

5133 

5296 

23 

3961 

4131 

4302 

4472 

4642 

5324 

5494 

5665 

24 

4267 

4444 

4622 

4800 

4978 

5689 

5867 

6044 

25 

4583 

4769 

4954 

5139 

5324 

6065 

6250 

6435 

26 

4911 

5104 

5296 

5489 

5681 

6452 

6644 

6837 

27 

5250 

5450 

5650 

5850 

6050 

6850 

7050 

7250 

28 

5600 

5807 

6015 

6222 

6430 

7259 

7467 

7674 

29 

5961 

6176 

6391 

6606 

6820 

7680 

7894 

8109 

30 

6333 

6556 

6778 

7000 

7'222 

8111 

8333 

8555 

31 

6717 

6946 

7176 

7406 

7635 

8554 

8783 

9013 

32 

7111 

7348 

7585 

7822 

8059 

9007 

9244 

9482 

33 

7517    7761 

8006 

825'0 

8494 

9472 

9717 

9962 

34 

7933    8185 

8437 

86S9 

8941 

9948 

10200 

10452 

35 

8361  I  8620 

8880 

9139 

9398 

10435 

10694 

10954 

36 

8800  |  9067    9333 

9600 

9867 

10933 

11200 

11467 

37 

9250 

9524 

9708 

1007'2 

10346 

11443 

11717 

11991 

38 

9711 

9993 

10274 

10556 

10837 

11963 

12244 

12526 

'39 

10183 

10472   10761 

11050 

11339- 

12494 

12783 

13072 

40 

10667 

10963   11259 

11556 

•11852 

13037 

13333 

13630 

41 

11161 

11465   11769 

12072 

12376 

13591 

13894 

14198 

42 

11667 

11978   1*289 

12600 

12911 

14156 

14467 

14778 

43 

12383 

12502   12820 

13139 

13457 

14731 

15050   15369 

44 

12711 

13037   133G3 

13689 

14015 

15319 

15644   15970 

45 

13250 

13583  !  13917 

14250 

14583 

15917 

16250  1  16583 

46 

13800 

14141 

14481 

11822 

15163 

16526 

16867 

17207 

47 

14361 

14709 

15057 

15406 

15754 

17146 

17494 

17843 

48 

14933 

15289 

15644 

16000 

16356 

17778 

18133 

18489 

49 

15517 

15880 

16243 

16606 

16968 

18420 

18783 

19146 

50 

16111 

16481 

16852 

17222 

17592 

19074 

19444 

19815 

51 

16717 

17094 

17472 

17850 

18228 

19739 

20117 

20494 

52 

17333 

17719 

18104 

18489 

18874 

20415 

20800 

21185 

53 

17961 

18354 

18746 

19139 

19531 

21102 

21494 

21887 

54 

18600 

19000 

19400 

19800 

20200 

21800 

22200 

22600 

55  . 

19250 

19657 

20065 

20472 

20880 

22509 

22917 

23324 

56 

19911 

20326 

20741 

21156 

21570 

23230 

23644 

24059 

57 

205&3 

21006 

21428 

21850 

22272 

23961 

243a3 

24805 

58 

21267 

21G96 

22126 

22556 

22985 

24704 

25133 

25563 

'    59 

21961 

22398 

2:2835 

23272 

23709 

25457 

25894 

26332 

|   ao 

22667 

23111 

23556 

24000 

24444 

26222 

26667 

27111 

347 


TABLE  XIV. — CUBIC  YARDS  PER  100  FEET.      SLOPES  2   ;  1. 


Depth 

Base 
12 

Base 
14 

Base 
16 

Base 
18 

Base 
20 

Ba-e 
28 

Base 
30 

Base 
32 

1 

52 

59 

67 

74 

81 

111 

119 

126 

2 

119 

133 

148 

163 

178 

237 

252 

267 

3 

200 

222 

244 

267 

289 

378 

400 

422 

4 

296 

326 

356 

385 

415 

5&3 

563 

693 

5 

407 

444 

481 

519 

556 

704 

741 

778 

6 

533 

578 

622 

637 

711 

889 

933 

978 

7 

674 

726 

778 

830 

881 

1089 

1141 

1193 

8 

830 

889 

948 

1007 

1067 

1304 

1363 

1422 

9 

1000 

1067 

1133 

1200 

1267 

1533 

1600 

1667 

10 

1185 

1259 

1333 

1407 

1481 

1778 

1852 

•1926 

11 

1385 

1467 

1548 

1630 

1711 

"  2037 

2119 

2200 

12 

1600 

1689 

1778 

1867 

1956 

2311 

2400 

2489 

13 

1830 

1926 

2022 

2119 

2215 

2600 

2696 

2793 

14 

2074 

2178 

2281 

2385 

2489 

2904 

3007 

8111 

15 

2333 

2444 

2556 

2667 

2778 

3222 

3333 

3444 

16 

2607 

2^6 

2844 

2963 

3081 

3556 

3674 

3793 

ir 

2896 

3022 

3148 

3274 

3400 

3904 

4030 

4156 

18 

3200 

o333 

3467 

3600 

3733 

4267 

4400 

4533 

19 

3519 

3659 

3800 

3941 

4081 

4644 

4785 

4926 

20 

3852 

4000 

4148 

4296 

4444 

5037 

5185 

5333 

21 

4200 

4356 

4511 

4667 

4822 

5444 

5600 

5756 

22 

4563 

4rso 

4889 

5052 

5215 

5867 

6030 

6193 

23 

4941 

5111 

5281 

545S 

5622 

6304 

6474 

6644 

24 

5333 

5511 

5689 

5867 

6014 

6756 

6933 

7111 

25 

5741 

5926 

6111 

6296 

6481 

7222 

7407 

7593 

26 

6163 

6356 

6548 

6741 

6933 

7704 

7896 

8089 

27 

6600 

6800 

7000 

7200 

7400 

8200 

8400 

8600 

28 

7052 

7259 

7467 

7674 

7881 

8711 

8919 

9126 

29 

7519 

77ii3 

7948 

8163 

8378 

9237 

9452 

9667 

30 

8000 

8222 

8444 

8667 

8389 

9778 

10000 

10222 

31 

8496 

8726 

8956 

9185 

9415 

10333 

10563 

10793 

32 

9007 

9244 

0481 

9719 

9956 

10904 

11141 

11378 

33 

9533 

9778 

10022 

10267 

10511 

11489 

11733 

11978 

34 

10074 

10326 

10578 

10830 

11081 

12089 

12341 

12593 

35 

10030 

10889 

11148 

11407 

11(567 

12704 

12963 

13222 

36 

11200 

11467 

11733 

12000 

12267 

13333 

13600 

13867 

37 

11785 

12059 

12333 

12607 

12.81 

13978 

14252 

14526 

38 

12385 

12667 

12948 

13230 

13511 

14637 

14919 

15200 

39 

13000 

13289 

13578 

13867 

14156 

15311 

15600 

15889 

40 

13630 

13926 

14222 

14519 

14815 

16000 

16296 

16593 

41 

14274 

14578 

14H81 

15185 

15489 

16704 

17007 

17311 

42 

14'.)33 

15244 

155L6 

15867 

10178 

17122 

17733 

18044 

43 

15607 

15926 

16224 

16563 

16881 

18156 

18474 

18793 

44 

16296 

16622 

16948 

17274 

17600 

18904 

19230 

19556 

45 

17000 

17333 

17667 

18000 

18333 

19667 

2001  :0 

20333 

46 

17719 

18059 

18400 

18741 

19081 

20444 

20785 

21126 

47 

18452 

18800 

19148 

19496 

19844 

21237 

21585 

21933 

48 

19200 

19556 

19911 

20267 

20622 

22044 

22400 

22756 

49 

19963 

20326 

20689 

21052 

21415 

22867 

23230 

23593 

50 

20741 

20711 

21481 

21852 

22222 

23704 

24074 

24444 

51 

2U33 

21911 

22289 

22667 

23044 

24556 

24933 

25311 

52 

22-341 

22726 

23111 

23496 

23881 

25422 

25807 

2619S 

53 

23163 

23556 

23948 

24341 

24733 

26304 

26696 

27089 

54 

24000 

24400 

24HOO 

2,V2<;0 

2">600 

27200 

27600 

28000 

55 

24852 

25259 

25667 

26074 

26481 

28111 

28519 

28926 

56 

25719 

26133 

26548 

26963 

27378 

29037 

29452 

29867 

57 

2d600 

27022 

27444 

27867 

28289 

29978 

30400 

30822. 

58 

27496 

27926 

28356 

28785 

29215 

30933 

31363 

31793 

59 

28407 

28844 

29281 

29719 

30156 

31904 

32341 

32778 

60 

29333 

29778 

30222 

30667 

31111 

32889 

33333 

33778 

348 


TABLE  XIV.— CUBIC  YARDS  PER  100  FEET.      SLOPES  8:1. 


Depth 

Base 

Base 

Base 

Base 

Base 

Base 

Base 

Base 

12 

14 

16 

18 

20 

28 

30 

32 

1 

56 

63 

70 

78 

85 

115 

122 

130 

2 

1:33 

148 

163 

178 

193 

252 

267 

281 

3 

233 

256 

278 

300 

322 

411 

433 

456 

4 

&56 

385 

415 

444 

474 

593 

622 

652 

5 

500 

537 

574 

611 

648 

796 

833 

870 

6 

667 

711 

756 

800 

844 

1022 

1067 

1111 

7 

856 

907 

959 

1011 

1063 

1270 

1322 

1374 

8 

1067 

1126 

1185 

1244 

1304 

1541 

1600 

1659 

9 

1300 

1367 

1433 

1500 

1567 

1833 

1900 

1967 

10 

1556 

1630 

1704 

1778 

1852 

2148 

2222 

2296 

11 

1833 

1915 

1996 

2078 

2159 

2485 

2567 

2648 

12 

-2133 

2222 

2311 

2400 

2489 

2844 

2933 

3022 

13 

2456 

2552 

2648 

2744 

2841 

3226 

3322 

3419 

14 

2800 

2904 

3007 

3111 

3215 

3630 

3733 

3837 

15 

3167 

3278 

3389 

3500 

3611 

4056 

4167 

4278 

16 

3556 

3674 

3793 

3911 

4030 

4504 

4622 

4741 

17 

3967 

4093 

4219 

4344 

4470 

4974 

5100 

5226 

18 

4400 

4533 

4667 

4800 

4933 

5467 

5600 

5733 

19 

4856 

4996 

5137 

5278 

5419 

5981 

6122 

6263 

20 

5333 

5481 

5630 

5778 

5926 

6519 

6667 

6815 

21 

5833 

5989 

6144 

6300 

6456 

7078 

7233 

7389 

22 

6356 

6519 

6681 

6844 

7007 

7659 

7822 

7985 

23 

6900 

7070 

7241 

7411 

7581 

8263 

8433 

8504 

24 

7467 

7644 

7822 

8000 

8178 

8889 

9067 

9144 

25 

8056 

8241 

8426 

8611 

8?'96 

9537 

9722 

9807 

26 

8667 

8859 

9052 

9244 

9437 

10207 

10400 

10593 

27 

9300 

9500 

9700 

9900 

10100 

10900 

11100 

11300 

28 

9956 

10163 

10370 

10578 

10785 

11615 

11822 

12030 

29 

10633 

10848 

11063 

11278 

11493 

12352 

12567 

12781 

30 

11333 

11556 

11778 

12000 

12222 

13111 

13333 

13556 

31 

12056 

12285 

12515 

12744 

12974 

13893 

14123 

14352 

32 

12800 

13037 

132?'4 

13511 

13748 

14696 

14933 

15170 

33 

13567 

13811 

14056 

14300 

14544 

15522 

15767 

16011 

34 

14356 

14607 

14859 

15111 

15-J63 

16370 

16622 

16874 

35 

1516? 

15426 

15685 

15944 

16204 

17241 

17500 

17759 

36 

16000 

16267 

16533 

16800 

17067 

18133 

18400 

18667 

37 

16856 

17130 

17404 

17678 

17952 

19048 

19322 

19596 

38 

17733 

18015 

18296 

18578 

18859 

19985 

20267 

20548 

39 

18633 

18922 

19211 

19500 

19789 

20944 

21233 

21522 

40 

19556 

19852 

20148 

20444 

20741  ' 

21926 

22222 

22516 

41 

20500 

20804 

21107 

21411 

21715 

22930 

23233 

2:3537 

4£ 

21467 

21778 

22089 

22400 

22711 

23956 

24267 

24578 

43 

22456 

22774 

23093 

23411 

23730 

25004 

25322 

25641 

44 

23467 

23793 

24119 

24444 

24770 

26074 

26400 

26726 

45 

21500 

24833 

25167 

25.500 

25833 

27167 

27500 

27833 

46 

25556 

25896 

26237 

26578 

26919 

28281 

28622 

28963 

47 

26633 

26981 

27330 

27678 

28026 

29419 

29767 

30115 

48 

27733   28089 

28444 

28800 

29156 

30578 

30933 

31289 

49 

2885G  I  29219 

20581 

2994-4 

30307 

31759 

32122 

32485 

50  . 

30000 

30370 

30741 

31111 

31481 

32963 

33333 

33704 

51 

31167 

31544 

31922 

32300 

32678 

34189 

34567 

34944 

52 

32356 

32741 

33126 

33511 

33896 

35437 

35822 

36207 

53 

33567 

33959 

343&i 

34744 

35137 

36707 

37100 

37493 

54 

34800 

35200 

35600 

36000 

36400 

38000 

38400 

38800 

55 

36056 

36463 

36870 

37278 

37685 

39315 

39722 

4(130 

56 

37333 

37748 

38163 

38578 

38993 

40652 

41067 

41481 

57 

38633 

39056 

39478 

39900 

40322 

42011 

42433 

42856 

58 

39956 

40385 

40815 

41244 

41674 

43393 

43822 

44252 

59 

41300 

41737 

4-2174 

42611 

43048 

44796 

45233 

45670 

60 

42667 

43111 

43556 

44000 

44444 

46222 

46667 

47111 

349 


TABLE  XV.— CUBIC  YARDS  IN  100  FEET  LENGTH. 


Area. 

& 

Cubic 
Yards. 

Area. 

f?: 

Cubic 
Yards. 

Area. 

9t 

Cubic 
Yards. 

Area. 

11 

Cubic 
Yards. 

Area. 

ft 

Cubic 
Yards. 

1 

3.7 

51 

188.9 

101 

374.1 

151 

559.3 

201 

744.4 

2 

7.4 

52 

192.6 

102 

377.8 

152 

563.0 

202 

748.2 

3 

11.1 

53 

19(5.3 

103 

381.5 

153 

500.7 

203 

.  751.9 

4 

14.8 

54 

200.0 

104 

385.2 

154 

570.4 

204 

755.6 

5 

18.5 

55 

203.7 

105 

388.9 

155 

574.1 

205 

759.3 

6 

22  2 

56 

207.4 

106 

392.6 

166 

577.8 

206 

763.0 

7 

25  '.9 

57 

211.1 

107 

396.3 

157 

581.5 

207 

766.7 

8 

29.6 

58 

214.8 

108 

400.0 

158 

585.2 

208 

770.4 

9 

33.3 

59 

218.5 

109 

403.7 

159 

588.9 

209 

774.1 

10 

37.0 

60 

2-22.2 

110 

407.4 

160 

592.6 

210 

777  .  8 

11 

40.7 

61 

225.9 

111 

411.1 

161 

596.3 

211 

781.5 

12 

44.4 

62 

2-^9.6 

112 

414.8 

162 

600.0 

212 

785.2 

13 

48.1 

63 

233.3 

113 

418.5 

163 

603.7 

213 

788.9 

14 

51.9 

64 

237.0 

114 

422.2 

164 

607.4 

214 

792.6 

15 

55.6 

65 

240.7 

\     115 

425.9 

165 

61  1  .  1 

215 

796.3 

16 

59.3 

66 

244.4 

116 

429.6 

166 

614  8 

216 

800.0 

17 

63.0 

67 

248.2 

117 

433.3 

167 

618.5 

217 

803.7 

18 

66.7 

68 

251.9 

118 

437.0 

168 

622.2 

218 

807.4 

19 

70.4 

69 

255.6 

119 

440.7 

169 

625.9 

219 

811.1 

20 

74.1 

70 

259.3 

120 

444  .4 

170 

629.6 

220 

814.8 

21 

77.8 

71 

263.0 

121 

448.2 

171 

633.3 

221 

818.5 

22 

81.5 

72 

266.7 

122 

451.9 

172 

637.0 

222 

822.2 

23 

85.2 

73 

270.4 

123 

455  6 

178 

640.7 

223 

825.9 

24 

88.9 

74 

274.1 

124 

459.3 

174 

644.4 

224 

829.6 

25 

92.6 

75 

277.8 

125 

403.0 

175 

648.2 

225 

833.3 

26 

96.3 

76 

281.5 

126 

466.7 

176 

651.9 

226 

837.0 

27 

100.0 

77 

285  2 

127 

470.4 

177 

655.6 

227 

840.7 

28 

103.7 

78 

288.9 

128 

474  1 

178 

659.3 

228 

844.4 

29 

107.4 

79 

292.6 

129 

477.8 

179 

003.0 

229 

848.2 

30 

111.1 

80 

296.3 

130 

481.5 

180 

666.7 

230 

851.9 

31 

114.8 

81 

300.0 

131 

485.2 

181 

670.4 

231 

855.6 

32 

118.5 

82 

303.7 

132 

488  9 

182 

674.1 

232 

859.3 

33 

122.2 

83 

307.4 

188 

492.6 

183 

677.8 

233 

863.0 

34 

125.9 

84 

311.1 

134 

496.3 

184 

681.5 

234 

866.7 

35 

129  6 

85 

314.8 

135 

500.0 

185 

685.2 

235 

870.4 

36 

133.3 

86 

318.5 

136 

503.7 

186 

688.9 

236 

874.1 

37 

137.0 

87 

322.2 

137 

507.4 

187 

692.6 

237 

877.8 

38 

140.7 

88 

325.9 

138 

511.1 

188 

696.3 

238 

881.5 

39 

144  4 

89 

329.6 

139 

514.8 

189 

700.0 

239 

885.2 

40 

148.2 

90 

333.3 

\     140 

518.5 

190 

703.7 

240 

888.9 

41 

151.9 

91 

337.0 

141 

522.2 

191 

707.4 

241 

892.6 

42 

155.6 

92 

340.7 

I     142 

525.9 

192 

711.1 

242 

896.3 

43 

159.3 

93 

344.4 

j     143 

529.6 

193 

714.8 

243 

900.0 

44 

103.0 

94 

348.2 

\     144 

533.3 

194 

718.5 

244 

903.7 

45 

166.7 

95 

351.9 

145 

537.0 

195 

722.2 

245 

907.4 

46 

170.4 

96 

355.6 

146 

540.7 

196 

725.9 

246 

911.1 

47 

174.1 

97 

359.3 

147 

544.4 

197 

729.6 

247 

914.8 

48 

177.8 

98 

363.0 

148 

548.2 

198 

733.3 

248 

918.5 

49 

181.5 

99 

366.7 

149 

551.9 

199 

737.0 

249 

9C2.2 

50 

185.2 

100 

370.4 

150 

555.6 

200 

740.7 

250 

925.9 

350 


TABLE  XV.— CUBIC  YARDS  IN  100  FEET  LENGTH. 


Area 
Sq. 
Ft. 

Cubic 
Yards. 

Area 

f£ 

Cubic 
Yards 

Area 

& 

Cubic 
Yards. 

Area 

& 

Cubic 
Yards. 

Area 
Sq. 

Ft. 

Cubic 
Yards. 

251 

929.6 

301 

1114.8 

351 

1300.0 

401 

1485.2 

451 

1670.4 

252 

933.3 

302 

1118.5 

352 

1303.7 

402 

1488.9 

452 

1674.1 

253 

937.0 

303 

1122.2 

353 

1307.4 

403 

1492.6 

453 

1677.8 

254 

940.7 

304 

1125.9 

354 

1311.1 

404 

1496.3 

454 

1681.5 

255 

944.4 

305 

1129.6 

355 

1314.8 

405 

1500.0 

455 

1685.2 

256 

948.2 

306 

1133.3 

356 

1318.5 

406 

1503.7 

456 

1688.9 

257 

951.9 

307 

1137.0 

357 

1322.2 

407 

1507.4 

457 

1692.6 

258 

955.8 

308 

1140.7 

358 

1325.9 

408 

1511.1 

458 

1696.3 

859 

959.3 

309 

1144.4 

359 

1329.6 

409 

1514.8 

459 

1700.0 

260 

963.0 

310 

1148.2 

360 

1333.3 

410 

1518.5 

460 

1703.7 

261 

9(56.7 

311 

1151.9 

361 

1337.0 

411 

1522  2 

461 

1707.4 

262 

970.4 

312 

1155.6 

362 

1340.7 

412 

1525.9 

462 

1711.1 

263 

974.1 

313 

1159.3 

363 

1:344.4 

413 

1529.6 

463 

1714.8 

264 

977.8 

314 

1163.0 

364 

1348.2 

414 

1533.3 

464 

1718.5 

265 

981.5 

315 

1166.7 

365 

1351.9 

415 

1537.0 

465 

1722.2 

26(5 

985.2 

316 

1170.4 

366 

1355.6 

416 

1540.7 

466 

1725.9 

267 

988.9 

317 

1174.1 

367 

1359.3 

417 

1544.4 

467 

1729.6 

268 

992.6 

318 

1177.8 

368 

1363.0 

418 

1548.2 

468 

1733.3 

269 

996.3 

319 

1181.5 

369 

1360.7 

419 

1551.9 

469 

1737.0 

270 

1000.0 

320 

1185.2 

370 

1370.4 

420 

1555.6 

470 

1740.7 

271 

1003.7 

321 

1188.9 

371 

1374.1 

421 

1559.3 

471 

1744.4 

272 

1007.4 

322 

1192.6 

372 

1377.8 

422 

1563.0 

472 

1748.2 

273 

1011.1 

323 

1196.3 

373 

1381.5 

423 

1566.7 

473 

1751.9 

274 

1014.8 

324 

1200.0 

374 

1385.2 

424 

1570.4 

474 

1755.6 

275 

1018.5 

325 

1203.7 

375 

1388.9 

425 

1574.1 

475 

1759.3 

276 

1022.2 

326 

1207.4 

376 

1392.6 

426 

1577.8 

476 

1763.0 

277 

1025.9 

327 

1211.1 

377 

1396.3 

427 

1581.5 

477 

1766.7 

278 

10-29.6 

328 

1214.8 

378 

1400.0 

428 

1585.2 

478 

1770.4 

279 

1033.3 

329 

1218.5 

379 

1103.7 

429 

1588.9 

479 

1774.1 

280 

1037.0 

330 

1222  2 

380 

1407.4 

430 

1592.6 

480 

1777.8 

281 

1040.7 

331 

1325.9 

381 

1411.1 

431 

1596.3 

481 

1781.5 

282 

1044.4 

332 

1229.6 

382 

1414.8 

432 

1600.0 

482 

1785.2 

283 

1048.2 

333 

1233.3 

383 

1418.5 

433 

1603.7 

4&3 

1788.9 

284 

1051.9 

334 

1237.0 

384 

1422.2 

434 

1607.4 

484 

1792.6 

285 

1055.6 

335 

1240.7 

385 

1425.9 

4-35 

1611.1 

485 

1796.3 

286 

1059.3 

336 

1244.4 

386 

1429  6 

436 

1614.8 

486 

1800.0 

287 

1063.0 

337 

1248.2 

387 

1433.3 

437 

1618.5 

487 

1803.7 

288 

1066.7 

338 

1251.9 

388 

1437.0 

438 

1622.2 

488 

1807.4 

289 

1070.4 

339 

1255.6 

389 

1440.7 

439 

1625.9 

489 

1811.1 

290 

1074.1 

340 

1259.3 

390 

1444.4 

440 

1629.6 

490 

1814.8 

291 

1077.8 

341 

1263.0 

391 

1448.2 

441 

1633.3 

491 

1818.5 

292 

1081.5 

342 

1266.7 

392 

1451.9 

442 

1637.0 

492 

1822.2 

293 

1085.2 

343 

1270.4 

393 

1455.6 

443 

1640.7 

493 

1825.9 

294 

1088.9 

344 

1274.1 

394 

1459.3 

444 

1644.4 

494 

1829.6 

295 

1092.6 

345 

1277.8 

395 

1463.0 

445 

1648  2 

495 

1833.3 

296 

1096.3 

346 

1281.5 

396 

1466.7 

446 

1651.9 

496 

1837.0 

297 

1100.0 

347 

1285.2 

397 

1470.4 

447 

1655.6 

497 

1840.7 

298 

1103.7 

348 

1288.9 

398 

1474.1 

448 

1659.3 

498 

1844.4 

299 

1107.4 

349 

1292.6 

399 

1477.8 

449 

1663.0 

499 

1818.2 

300 

1111.1 

350 

1296.3 

400 

1481.5 

450 

1666.7 

500 

1851.9 

351 


TABLE  XV.— CUBIC  YARDS  IN  100  FEET  LENGTH. 


Area. 

St 

Cubic 
Yards. 

Area. 

1?: 

Cubic 
Yards. 

Area. 

It 

Cubic 
Yards. 

Area. 

11 

Cubic 
Yards. 

Area. 

11 

Cubic 
Yards. 

501 

1855.6 

551 

2040.7 

601 

2225.9 

651 

2411.1 

701 

2596.3 

502 

1859.3 

552 

2044.4 

602 

2229.6 

652 

2414.8 

702 

2600.0 

503 

1863.0 

553 

2048.2 

603 

2233.3 

653 

2418.5 

703 

2603.7 

504 

1866.7 

554 

2051.9 

604 

2237.0 

654 

2422.2 

704 

2607.  4 

505 

1870.4 

555 

2055.6 

605 

2240.7 

655 

2425.9 

705 

2611.1 

506 

1874.1 

556 

2059.3 

606 

2244.4 

656 

2429.6 

706 

2614.8 

507 

1877.8 

557 

2063.0 

607 

2248.2 

657 

2433.3 

707 

2618.5 

508 

1881.5 

558 

2066.7 

608 

2251.9 

658 

2437.0 

708 

2622.2 

509 

1885.2 

559 

2070.4 

609 

2255.6 

659 

2440.7 

709 

2(525.9 

510 

1888.9 

560 

2074.1 

610 

2259.3 

660 

2444.4 

710 

2629.6 

511 

1892.6 

561 

2077.8 

611 

2263.0 

661 

2448.2 

711 

2633.3 

512 

1896.3 

562 

2081.5 

612 

2266.7 

662' 

2451.9 

712 

2637.0 

513 

1900.0 

563 

2085.2 

613 

2270.4 

663 

2455.6 

713 

2640.7 

514 

1903.7 

564 

2088.9 

614 

2274.1 

664 

2459.3 

714 

2644.4 

515 

1907.4 

565 

2092.6 

615 

2277.8 

665 

2463.0 

715 

2648.2 

516 

1911.1 

566 

2096.3 

616 

2281.5 

666 

2466.7 

716 

2651.9 

517 

1914.8 

567 

2100.0 

617 

2285.2 

667 

2470.4 

717 

2655.6 

518 

1918.5 

568 

2103.7 

618 

2288.9 

668 

2474.1 

718 

2659.3 

519 

1922.2 

569 

2107.4 

619 

2292.6 

669 

2477.8 

719 

2663.0 

520 

1925.9 

570 

2111.1 

620 

2296.3 

670 

2481.5 

720 

2666.7 

521 

1929  6 

571 

2114.8 

621 

2300.0 

671 

2485.2 

721 

2670.4 

522 

1933.3 

572 

2118.5 

622 

2303.7 

672 

2488.9 

722 

2674.1 

523 

1937.0 

573 

2122.2 

623 

2307.4 

673 

2492  6 

723 

2677.8 

524 

1940.7 

574 

2125.9 

624 

2311.1 

674 

2496.3 

724 

2C81.5 

525 

1944.4 

575 

2129.6 

625  \  2314.8 

675 

2500.0 

725 

2685.2 

526 

1948.2 

576 

2133.3 

626 

2318.5 

676 

2503.7 

726 

2(188.9 

527 

1951.9 

577 

2137.0 

627 

2322.2 

677 

2507.4 

727 

2(i92.6 

528 

1955.6 

578 

2140.7 

628 

2325.9 

678 

2511.1 

728 

2(1%.  3 

529 

1959.3 

579 

2144.4 

629  !  2329.6 

679 

2514.8 

729 

2700.0 

530 

1963.0 

580 

2148  2 

630      2333.3 

680 

2518.5 

730 

2703.7 

531 

1966.7 

581 

2151.9 

631 

2337.0 

681 

2522.2 

731 

2707.4 

532 

1970.4 

582 

2155.6 

632 

2340.7 

682 

2525.9 

732 

2711.1 

533 

1974.1 

583 

2159.3 

633 

2344.4 

683 

2529.6 

733 

2714.8 

534 

1977.8 

584 

2163.0 

634 

2348.2 

684 

2533.3 

734 

2718.5 

535 

1981.5 

585 

2166.7 

635 

2351.9 

685 

2537.0 

735 

2722.2 

536 

1985.2 

586 

2170.4 

636 

2355.6 

686 

2540  7 

736 

2725.9 

537 

1988.9 

587 

2174.1 

637 

2359.3 

687 

2544.4 

737 

2729.6 

538 

1992.6 

588 

2177.8 

638 

2363  0 

688 

2548.2 

738 

2733.3 

539 

1996.3 

589 

2181.5 

639 

2366.7 

689 

2551.9 

739 

2737.0 

540 

2000.0 

590 

2185.2 

640 

2370.4 

690 

2555.6 

740 

2740.7 

541 

2003.7 

591 

2188.9 

641 

2374.1 

691 

2559.3 

741 

2744.4 

542 

2007.4 

592 

2192.6 

642 

2377.8 

692 

2563.0 

742 

2748.2 

543 

2011.1 

593 

2196.3 

643 

2381.5 

693 

2566.7 

743 

2751.9 

544 

2014.8 

594 

2200.0 

644 

2385.2 

694 

2570.4 

744 

2755.6 

545 

2018.5 

595 

2203.7 

645 

2388.9 

695 

2574.1 

745 

2759.3 

546 

2022.2 

596 

2207.4 

646 

2392.6 

696 

2577.8 

746 

2763.0 

547 

2025.9 

597 

2211.1 

647 

2396.3 

697 

2581.5 

747 

2766.7 

548 

2029.6 

598 

2214.8 

648. 

2400.0 

698 

2585.2 

748 

2770.4 

549 

2033.3 

599 

2218.5 

649 

2403.7 

699 

2588.9 

749 

2774.1 

550 

2037.0 

600 

2222.2 

650 

2407.4 

700 

2592.6 

750 

2777.8 

352 


TABLE  XV.-CUBIC  YARDS  IN  100  FEET  LENGTH. 


Area. 

ft 

Cubic 
Yards. 

Area. 
Sq. 
K, 

Cubic 
Yards. 

Area. 

£ 

Cubic 
Yards. 

Area. 

11: 

Cubic 
Yards. 

Area. 

ft 

Cubic 
Yards. 

751 

2781.5 

801 

2966.7 

851 

3151.9 

901 

3337.0 

951 

3522.2 

752 

2785.2 

802      2970.4 

852 

3155.6 

902 

3340.7 

952 

3525.9 

753 

2788.9 

803 

2974.1 

853 

3159.3  : 

903 

3344.4 

953 

3529.6 

754 

2792.6 

804 

2977.8 

854 

3163.0 

904 

3348.2 

954 

3533.3 

755 

2796.3 

805      2981.5 

!     855 

3166.7 

905 

3351.9 

955 

3537.0 

756 

2800.0 

806 

2985.2 

856 

3170.4 

906 

3355.6 

956 

3540.7 

757 

2803.7 

807 

2988.9 

857 

3174.1 

907 

3359.3 

957 

3544.4 

758 

2807.4 

808 

2992.6 

858 

3177.8 

908 

3363.0 

958 

3548.2 

759 

2811.1 

809 

2996.3 

859 

3181.5 

909 

3366.7 

959 

3551.9 

760 

2814.8 

810 

3100.0 

860 

3185.2  : 

910 

3370.4 

960 

3555.6 

761 

2818.5 

811 

3003.7 

861 

3188.9 

911 

3374.1 

961 

3559.3 

762 

2822.2 

812 

3007.4 

i     862 

3192.6 

912 

3377.8 

962 

3563.0 

763 

2825.9 

813 

3011.1 

863 

3196.3 

913 

3381.5 

963 

3566.7 

764 

2829.6 

814 

3014.8 

864 

3200.0 

914 

3385.2 

964 

3570.4 

765 

2833  3 

815 

3018.5 

865 

3203.7 

915 

3388.9 

965 

3574.1 

766 

2837.0 

816 

3022.2 

866 

3207.4 

916 

3392.6 

966 

3577.8 

767 

2840.7 

817 

3025.9 

867 

3211.1 

917 

3396.3 

967 

3581.5 

768 

2844.4 

818 

3029.6 

868 

3214.8 

918 

3400.0 

968 

3585.2 

769 

2848.2 

819 

3033.3 

869 

3218.5 

919 

3403.7 

969 

3588.9 

770 

2851.9 

820 

3037.0 

870 

3222.2 

920 

3407.4 

970 

3592.6 

771 

2855.6 

821 

3040.7 

871 

3225.9 

921 

3411.1 

971 

3596.3 

772 

2859.3 

822 

3044.4 

872 

3229.6 

922 

3414.8 

972 

3600.0 

773 

2863.0 

823 

3048.2 

873 

3233.3 

923 

3418.5 

973 

3603.7 

774 

2866.7 

824 

3051.9 

;    874 

3237.0 

924 

3422.2 

974 

3607.4 

775 

2870.4 

825 

3055.6 

I     875 

3240.7 

925 

3425.9 

975 

3611.1 

776 

2874.1 

826 

3059.3 

876 

3244.4 

926 

3429.6 

976 

3614.8 

777 

2877.8 

827 

3063.0 

1    877 

3248.2 

927 

3433.3 

977 

3618.5 

778 

2881.5 

828 

3066.7 

<    878 

3251.9 

928 

3437.0 

978 

3622.2 

779 

2885.2 

829 

3070.4 

\    879 

3255  6   \ 

929 

3440.7 

979 

3625.9 

780 

2888.9 

830 

3074.1 

880 

3259.3   ! 

930 

3444.4 

980 

3629.6 

781 

2892.6 

831 

3077.8 

881 

3263.0   j 

931 

3448.2 

981 

3633.3 

782 

2896.3 

832 

3081.5 

i     882 

3266.7 

932 

3451.!) 

982 

3637.0 

783 

2900.0 

833 

3085.2 

j     883 

3270.4 

933 

3455.6 

983 

3640.7 

784 

2903.7 

834 

3088.9 

884 

3274.1 

934 

3459.3 

984 

3644.4 

785 

2907.4 

835 

3092.6 

885 

3277.8 

935 

3463.0 

985 

3648.2 

786 

2911.1 

836 

3096.3 

i     886 

3281.5 

936 

3466.7 

986 

3651.9 

787 

2914.8 

837 

3100.0 

;     887 

3285.2 

937 

3470.4 

987 

3655.6 

788 

2918.5 

838 

3103.7 

888 

3288.9 

938 

3474.1 

988 

3659.3 

789 

2922.2 

839 

3107.4 

1     889 

3292.6 

939 

3477.8 

989 

3663.0 

790 

2925.9 

840 

3111.1 

890 

3296.3 

940 

3481.5 

990 

3666.7 

791 

2929.6 

841 

3114.8 

891 

3300.0 

941 

3485  2 

991 

3670.4 

792 

2933.3 

842 

3118.5 

892 

3303.7  : 

942 

3488.9 

992 

3674.1 

793 

2937.0 

843 

3122.2 

893 

3307.4   ' 

943 

3492.6 

993 

3677.8 

794 

2940.7 

844 

3125  9 

:       894 

3311.1 

944 

3496.3 

994 

3681.5 

795 

2944.4 

845 

3129.6 

895 

3314.8  , 

945 

3r,oo.o 

995 

3685.2 

796 

2948.2 

846 

3133.3 

896 

3318.5 

946 

3503.7 

996 

3688.9 

797 

2951.9 

847 

3137.0 

i     897 

3322.2 

947 

3507.4 

997 

3692.6 

798       2955.6 

848 

3140.7 

!     898 

3325.9 

948 

3511.1 

998 

3696.3 

799 

2959.3 

849 

3144.4 

i     899 

3329.6 

949 

3514.8 

999 

3700.0 

800 

2963.0 

850 

3148.2 

!    900 

3333.3 

950 

3518.5 

1000 

3703.7 

353 


TABLE  XVI. 


CONVERSION  OF  ENGLISH  INCHES  INTO  CENTIMETRES. 

Ins. 

0 

1 

2 

3 

4 

5 

6 

rt 

8 

9 

Cm. 

Cm. 

Cm. 

Cm. 

Cm. 

Cm. 

Cm. 

Cm. 

Cm. 

Cm. 

0 

0.000 

2.540      5.080 

7.620 

10.16 

12.70 

15.24 

17.7.8 

20.32 

22.86 

10 

25.40 

27.94      30.48 

33.02 

35.56 

38.10 

40.64 

43.18 

45.72 

48.26 

20 

50.80 

53.34J     55.88 

58.42 

60.96 

63.50 

66.04 

68.58 

71.12 

73.66 

30 

76.20 

78.74!     81.28 

83.82 

86  36 

88.90 

91.44 

93.98 

96.52 

99.06 

40 

101.60 

104.14!   106.68 

109.22 

111.76 

114.30 

116.84  119.38 

121.92 

124.46 

50 

127.00 

129.54    132.08 

134.62 

137.16 

139.70 

142.24 

144.78 

147.32 

149.86 

60 

152.40 

154.94    157.48 

160.02 

162.56 

165.10 

167.64  170.18 

172.72 

175.26 

70 

177.80 

180.34!  182.88 

185.42 

187.96 

190.50 

193.04 

195.58 

198.12 

200.96 

80 

203.20 

205.74    208.28 

210.82 

213.36 

215.90 

218.44 

220.98 

223.52 

226.06 

90 

228.60 

231.  14i  233.68 

236.22 

238.76 

241  .  30 

243.  84  1246.38 

248.92 

251.46 

100 

254.00 

256.  54  ;  259.08 

261.62 

264  10 

266.70 

269  241271.78  274.  32 

27'(>.8'.i 

CONVERSION  OF  CENTIMETRES  INTO  ENGLISH  INCHES. 

Cm. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins.      Ins.      Ins. 

Ins. 

Ins. 

0 

0.000 

0.394 

0.787;     1.181 

1.575 

1.969    2.362    2.756 

3.150    3.543 

10 

3.937 

4.331 

4.742     5.118 

5.512 

5.906    6.299    6.693 

7.087i  7.480 

20 

7.874 

8.268 

8.662     9.055 

9.449 

9.843  10.236  10.630  11.024  ill.  418 

30 

11.811 

12.205 

12.599    12.992 

13.386 

13.  780  14.  1V3  14.  567114.  961115.  855 

40 

15.748 

16.142 

16.530    16.929 

17.323 

17.  71718.  Ill  18.504,18.898  19.292 

50 
60 

19.685 
23.622 

20.079    20.473    20.867 
24.016!  24.410    24.804 

21.260 
25.197 

21.654122.04822  441  122.835|23.229 
25  .  591  25  .  985  26  .  378  -J(j  .  772  27  .  1  66 

70 

27.560 

27.953,  28.347    28.741 

29.134 

29  .  528  29  .  922  30  .  3  1  6  30  .  701)  31  .  103 

80 

31.497 

31.890    82.284    32.678 

33.071 

33.465  33.859  34.253  34.646  35.040 

90 

35.434 

35.827!  36.221    36.615 

37.009 

37  .  402  37  .  796  38  .  1  90  38  .  583  38  .  977 

100 

39.370)  39.7641  40.158    40.552 

40.945 

41.33941.73342.126 

42.52042.914 

CONVERSION  OF  ENGLISH  FEET  INTO  METRES. 

Feet. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Met. 

Met. 

Met. 

Met. 

Met. 

Met. 

Met. 

Met. 

Met. 

Met. 

0 

0.000 

0.3048 

0.6096 

0.9144 

1.2192 

1.5239 

1.82872.1335 

2.4383 

2.7431 

10 

3.0479 

3.3527 

3.6575 

3.9623 

4.2671 

4.5719 

4.87675.1815 

5.4863 

5.7911 

20 

6.0359 

6.4006 

6.7055 

7.0102 

7.3150 

7.61987.92468.2294 

8.5342 

8.8390 

30 

9.1438 

9.4486 

9.7534 

10.058 

10.363 

10.66810.97211.277 

11.582 

11.887 

40 

12.192 

12.496 

12.801 

13.106 

13.411 

13.71614.02014.325 

14.630 

14.935 

50 

15.239 

15.544 

15.849 

16.154 

16.459 

16.763  17.06817.373 

17.678 

17.983 

60 

18.287 

18.592 

18.897 

19.202 

19.507 

19.811J20.11620.421 

20.726 

21.031 

70 

21.335 

21.640 

21.945 

22.250 

22.555 

22.85923.16423.46'.) 

23  774 

24.079 

80 

24.383 

24.688 

24.993 

25.298 

25.602 

25.907:26.21226.517 

26.82227.126 

90 

27.431 

27.736 

28.041 

28.346 

28.651 

28.  955  '29.  260  29.  565 

29.  870130.  174 

100 

30.479    30.784 

31.089 

31.394 

31.698    3200332.30832.613 

32.  91  8  133.222 

CONVERSION  OF  METRES  INTO  ENGLISH  FEET. 

Met. 

JL_LJ_ 

*       » 

4 

5 

6 

7 

8 

9 

Feet.     Feet. 

Feet.     Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

0 

0.000    3.2809 

6.5618    9.8427 

13.123 

16.404 

19.685 

22.906 

26.247 

29.528 

10 

32.809    36.090 

39.371    42.651 

45  .  932 

49.21352.494 

55.775 

59.056 

62:337 

20 

65.618    68.899 

72.179    75.461 

78.741 

82.02285.303 

88.584 

91.865 

95.146 

30 

98.427!  101.71 

104.99    108.27 

111.55 

114  831118.11 

121.39 

124.67 

127.96 

40 

131.24i  134.52 

137.80    141.08 

144.36 

147.64  150  92 

154.20 

157.48 

160.76 

50 

164.041  167.33 

170  61    173.89 

177.17 

180.45183.73 

187.01 

190.29 

193.57 

60 

196.85    200.13 

203.42    206.70 

209.98 

213.26216.54 

219.82 

223.10 

226.38 

70 

229.66    232.94 

236.22    239.51 

242.79 

246.07249.35 

252  63 

255.91 

259.19 

80 

262.47 

265.75 

269.03    272.31 

27'5.60 

278.88:282.16 

285.44 

288  72 

292.00 

-  90 

295.28 

298.56 

391.84    305.12 

308.40 

311.69314.97 

518.25 

321.53 

324.81 

100 

328.09    331.37 

334.65!  337.93 

341.21 

344.49;347.78 

351.06 

354.34  357.62 

354 


TABLE  XVH. 


CONVERSION  OF 

ENGLISH   STATUTE-MILES  INTO  KILOMETRES. 

Miles.  |     0 

1 

2 

3 

4 

5 

6 

T      1      S 

9 

Kilo. 

Kilo. 

Kilo. 

Kilo. 

Kilo.     Kilo. 

Kilo. 

Kilo. 

Kilo. 

Kilo. 

0 

0.0000 

1.6093 

3.2186 

4.82796.4372    8.0465 

9.6558  11.2652 

12.8745 

14.4818 

10 

16.093 

17.702 

19.312 

20.921 

22  530    24.139 

25.749J  27.358 

28.967 

30.577 

20 

32.186  33.795  35.405  37.014 

38.623    40.232 

41.8421  43  451 

45.060 

46.670 

30 

48.279  49.88851.498 

53.107 

54.716;  56.325 

57.935 

59.544 

61.153 

62.763 

40 

64.372 

65.981 

67.591 

69.200 

70.809 

72.418 

74.028 

75.637 

77.246 

78.856 

50 

80.46582.07483.684 

85.293 

86.902 

88.511 

90.121 

91  730 

93.339 

94.949 

60 

96.558 

98.167 

99  777 

101.39 

102.99 

104.60 

106  21 

107.82 

109.43 

111.04 

70 

112.65 

114.26115.87 

117.48 

119.08 

120.69 

122.801  123.91 

125.52 

127.13 

80 

128  74 

130.35 

131.96 

133  57 

135.17 

136.78 

138.39    140.00 

141.61 

143.22 

90 

144.85146.44 

148.05 

149.66 

151.26 

152.87 

154.48    156.09 

157.70!  159.31 

100 

160.93162.53164  14  165  75 

167.35 

168.96 

170.57    172.18 

173.79!  175.40 

t  \r\  vr\7  L^  OO>T/^  XT    /~^TJ*    T/"TT  r\  vr  T7<Tn  D  TT<  o    TXTT'/~\    TPXT/^ir  TCJTT    OT*  A  T^TTTTF  TVTTT  fo 

CONVERSION   OF   KILOMETRES  INTO   ENGLISH   STATUTE-MILES. 


Kilom.       0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Miles. 

Miles. 

Miles 

Miles. 

Miles 

.   Miles. 

Miles. 

Miles. 

Miles. 

Miles. 

0       0.0000 

0.6214 

1  2427 

1.8641 

2.485 

5    3.  1069 

3.7282 

4.3497 

4.9711 

5.5924 

10       6.2138 

6.8:352 

7.4565 

8.0780 

8.699 

4    9.3208 

9.9421 

10.562 

11.185 

11.  80S 

20        12.427 

13  049 

13  670 

14.292 

14.91 

3    15.534 

16.156 

16.776 

17.399 

18.01E 

30        18  641 

19.263 

19.884 

20.506 

21.12 

7    21.748 

23  370 

22.990 

23.613 

24.23S 

40       24.855 

25.477 

26.098 

26.720 

27.34 

1    27.962 

28.584 

29.204 

29.827 

30.447 

50       31.0G9 

31.690 

32.311 

32.933 

33.55 

4    34.175 

34.797 

35.417 

36.040 

36.66C 

60       37.282 

37.904 

38.525 

39  147  :  39  76 

8    40.389 

41.011 

41.631 

42.254 

42.874 

70       43.497 

44.118 

44.739 

45.361 

45.98 

2    46.603 

47.225 

47.845 

48.468 

49.08* 

80       49.711 

50.332 

00.9.-.3 

51.575 

52.19 

6    52.817 

53.439 

54.059 

54.682 

55.305 

90       55.924 

56.545 

57.166 

57.788 

58.40 

9    59  .  030 

59.652 

60.272 

60.895 

61.51E 

100       62.138 

62.759 

63.380 

64.002 

64.62 

•3    65.244 

65.866 

66.486 

67.109J  67.721 

TABLE   XVIII. 

LENGTH  IN  FEET  OF 

1'  ARCS  OF  LATITUDE 

AND  LONGITUDE. 

Lat. 

1'  Lat. 

1'  Long. 

Lat, 

1'  Lat. 

V  Long. 

1° 

6045 

6085 

31° 

6061 

5222 

2° 

6045 

6083 

32° 

6062 

J166 

3° 

6045 

6078 

33° 

6063 

3109 

4° 

6045 

6071 

34° 

6064 

5051 

5° 

6045 

6063 

35° 

6005 

1991 

6° 

6045 

6053 

36° 

6066 

4930 

7° 

6046 

6041 

37° 

6067 

1867 

8° 

6046 

6027 

38° 

6068 

4802 

9° 

6046 

6012 

39° 

6070 

4736 

10° 

6047 

5994 

40" 

6071 

4669 

11° 

6047 

5975 

41° 

6072 

1600 

J.)0 

6048 

5954 

42° 

6073 

1530 

13° 

6048 

5931 

43° 

6074 

4458 

14° 

6049 

5907 

44° 

6075 

1385 

15° 

6049 

5880 

45° 

6076 

4311 

16° 

6050 

5852 

46° 

6077 

1235 

17° 

6050 

5822 

47° 

- 

6078 

4158 

18° 

6051 

5790 

48° 

6079 

1080 

19° 

6052 

5757 

49° 

6080 

1001 

20° 

6052 

5721 

50° 

6081 

3920 

21  c 

6053 

5684 

51° 

6082 

3838 

22° 

6(54 

5646 

52° 

6084 

3755 

23° 

6054 

5605 

53° 

6085 

3671 

24° 

6055 

556'3 

54° 

6086 

3586 

25° 

6056 

5519 

55° 

6087 

3499 

26° 

6057 

5474 

56° 

6088 

3413 

27° 

6058 

5427 

57° 

6089 

3323 

28° 

6059 

5378 

58° 

6090 

3233 

29° 

6060 

5327 

59° 

6091 

3142 

30° 

6061 

5275 

60° 

6092 

3051 

355 


TABLE  XIX.— TO  REDUCE    MEAN  TO  SIDEREAL  TIME. 


Solar 

Add 

Solar 

Add 

Solar 

Add 

Solar 

Add 

Solar 

Add 

Hours. 

Min.  Sec. 

Min. 

Sec. 

Min. 

Sec. 

Sec. 

Sec. 

Sec. 

Sec. 

1 

0    9.86 

1 

0.16 

31 

5.09 

1 

0.00 

31 

0.08 

2 

0  19.71 

2 

0.33 

32 

5  26 

2 

0.01 

32 

0.09 

8 

0  29.57 

3 

0.49 

33 

5.42 

3 

0.01 

33 

0.09 

4 

0  39.43 

4 

0.66 

34 

5.59 

4 

0.01 

34 

0.09 

.      5 

0  49.28 

5 

0.82 

35 

5.75 

5 

0  01 

35 

0.10 

6 

0  59.14 

6 

0.99 

36 

5.92 

6 

0.02 

36 

0.10 

7 

1     9.00 

7 

1.15 

37 

6.08 

7 

0.02 

37 

0.10 

8 

'   1  18.85 

8 

1.31 

38 

6.24 

8 

0.02 

38 

0.10 

9 

1  28.71 

9 

1.48 

39 

6.41 

9 

0.03 

39 

0.11 

10 

1  38.57 

10 

1.64 

40 

6.57 

10 

0.03 

40 

0.11 

11 

1  48.42 

11 

1.81 

41 

6.74 

11 

0.03 

41 

0.11 

12 

1  58.28 

12 

1.97 

42 

6.90 

12 

0.03 

42 

0.12 

13 

2    8.13 

13 

2.14 

43 

7.07 

13 

0.04 

43 

0.12 

14 

2  17.99 

14 

2.30 

44 

7.23 

14 

0.04 

44 

0.12 

15 

2  27.85 

15 

2.46 

45 

7.39 

15 

0.04 

45 

0.12 

16 

2  37.70 

16 

2.63 

46 

7.56 

16 

0.04 

46 

0.13 

17 

2  47.56 

17 

2.79 

47 

7  72 

17 

0.05 

47 

0.13 

18 

2  57.42 

18 

2.96 

48 

7  '.89 

18 

0.05 

48 

0.13 

19 

3    7.27 

19 

3.12 

49 

8.05 

19 

0.05 

49 

0.13 

20 

3  17.13 

20 

3.29 

50 

8.22 

20 

0.06 

50 

0.14 

21 

3  26.99 

21 

3.45 

51 

8.38 

21 

0.06     I!     51 

0.14 

22 

3  36.84 

22 

3.61 

52 

8.54 

22 

0.06      1     52 

0.14 

28 

3  46.70 

23 

3.78 

53 

8.71 

23 

0.06     II     53 

0.15 

24 

3  56.56 

24 

3.94 

54 

8.87 

24 

0.07          54 

0.15 

25 

4    6.40 

25 

4.11 

55 

9.04 

25 

0.07     I'     55 

0.15 

26 

4  16.26 

26 

4.27 

56 

9.20 

26 

0.07      j     56 

0.15 

27 

4  26.13 

27 

4.44 

57 

9.37 

27 

0.08 

57 

0.16 

28 

4  36.00 

28 

4.60 

58 

9.53 

28 

0.08 

58 

0.16 

29 

4  45.86 

29 

4.76 

59 

9.69 

29 

0.08 

59 

0.16 

30 

4  55.71 

30 

4.93 

60 

9.86 

30 

0.08          60 

0.16 

356 


TABLE  XIX— Continued.-TO  REDUCE  SIDEREAL  TO  MEAN  TIME. 


Sid. 
Hours. 

Subtract 
Min.  Sec. 

Sid. 
Min. 

Sub- 
tract 
Sec. 

Sid. 
Min. 

Sub- 
tract 
Sec. 

Sid. 
Sec. 

Sub- 
tract 
Sec. 

Sid. 
Sec. 

Sub- 
tract 
Sec. 

1 

o  9.  as 

1 

0.16 

!     31 

5.08 

1 

0.00 

31 

0.08 

2 

0  19.66 

2 

0.33 

32 

5.24 

2 

0.01 

32 

0.09 

3 

0  29.49 

3 

0.49 

33 

5.41 

3 

0.01 

33 

0.09 

4 

0  39.32 

4 

0.66 

34 

5.57 

4 

0.01 

34 

0.09 

5 

0  49.15 

5 

0.82 

i    35 

5.73 

5 

0.01 

35 

0.10 

6 

0  58.98 

6 

0.98 

36 

5.90 

6 

0.02 

36 

0.10 

7 

8.81 

7 

.15 

37 

6.06 

7 

0.02 

37 

0.10 

8 

18.64 

8 

.31 

38 

6.23 

8 

0.02 

38 

0.10 

9 

28.47 

9 

.47 

39 

6.39 

9 

0.03 

39 

0.11 

10 

38.30 

10 

.64 

40 

6.55 

10 

0.03 

40 

0.11 

11 

48.12 

11 

.80 

41 

6.72 

11 

0.03 

41 

0.11 

12 

57.95 

ia 

.97 

42 

6.88 

12 

0.03 

42 

0.11 

13 

2    7.78 

13 

2.13 

43 

7.04 

13 

0.04 

43 

0.12 

14 

2  17.61 

14 

2.29 

!    44 

7.21 

14 

0.04 

44 

0.12 

15 

2  27.44 

15 

2.46 

45 

7.37 

15 

0.04 

45 

0.12 

16 

2  37.27 

16 

2.62 

46 

7.54 

16 

0.04 

46 

0.13 

1? 

2  47.10 

17 

2.79 

!     47 

7.70 

17 

0  05 

47 

0.13 

18 

2  56.93 

18 

2.95 

48 

7.86 

18 

0.05 

48 

0.13 

19 

3    6.76 

19 

3.11 

49 

8.03 

19 

0.05 

49 

0.13 

20 

3  16.59 

20 

3.28 

50 

8.19 

20 

0.06 

50 

0.14 

21 

3  26.42 

21 

3.44 

51 

8.36 

21 

0.06 

51 

0.14 

22 

3  36.25 

22 

3.60 

52 

8.52 

22 

0.06 

52 

0.14 

23 

3  46.08 

23 

3.77 

;    53 

8.68 

23 

0.06 

53 

0.14 

24 

3  55.91 

24 

3.93 

54 

8.85 

24 

0.07 

54 

0.15 

25 

4    5.74 

25 

4.10 

55 

9.01 

25 

0.07 

55 

0.15 

26 

4  15.57 

26 

4.26 

56 

9.17 

26 

0.07 

56 

0.15 

27 

4  25.41 

27 

4.42 

57 

9.34 

27 

0.07 

57 

0.16 

28 

4  35.24 

28 

4.59 

58 

9.50 

28 

0.08 

58 

0.16 

29 

4  45.07 

29 

4.75 

59 

9.67 

29 

0.08 

59 

0.16 

30 

4  54.90 

30 

4.92 

60 

9.83 

30 

0.08 

60 

0.16 

TABLE  XX. 


TO  REDUCE  DEGREES  TO  TIME. 

TO  REDUCE  TIME  TO  DEGREES. 

0 

H.  M. 

o 

H.  M. 

crl 

•  $ 

| 

M. 

0     / 

M. 

0     / 

' 

M.S. 

' 

M.S. 

tiO 

|l 

1 

1 

S. 

/   // 

S. 

/   n 

" 

S.  T. 

II 

S.  T. 

<u 
fi 

o  .— 

ffi  S 

;    O 

|  ffi 

1 

T. 

//  /// 

T. 

~7T77, 

1 

0   4 

51 

3  24 

101 

6  44 

1 

15 

1 

0  15 

51 

12  45 

2 

0   8 

52 

3  28 

102 

6  48 

n 

22i 

2 

0  30 

52 

13  0 

3 

0  12 

53 

3  32 

103 

6  52 

2 

30 

3 

0  45 

53 

13  15 

4 

0  16 

54 

3  36 

104 

6  56 

2* 

37* 

4 

1  0 

54 

13  30 

5 

0  20 

55 

3  40 

105 

7  0 

3 

45 

5 

1  15 

55 

13  45 

6 

0  24 

56 

3  44 

106 

7  4 

3* 

52* 

6 

1  30 

56 

14  0 

7 

0  28 

57 

3  48 

107 

7  8 

4 

60 

7 

1  45 

57 

14  15 

8 

0  32 

58 

3  52 

108 

7  12 

4* 

67* 

8 

2  0 

58 

14  30 

9 

0  36 

59 

3  56 

109 

7  16 

5 

75 

9 

2  15 

59 

14  45 

10 

0  40 

60 

4   0 

110 

7  20 

5* 

82| 

10 

2  30 

60 

15  0 

11 

0  44 

61 

4   4 

115 

7  40 

6 

90 

11 

2  45 

61 

15  15 

12 

0  48 

62 

4   8 

120 

8  0 

6| 

97.! 

12 

3  0 

6'^ 

15  30 

13 

0  52 

63 

4  12 

125 

8  20 

7 

105 

13 

3  15 

63 

15  45 

14 

0  56 

64 

4  16 

130 

8  40 

7* 

112* 

14 

3  30 

64 

16  0 

15 

1   0 

65 

4  20 

135 

9  0 

8 

120 

15 

3  45 

65 

16  15 

16 

1   4 

66 

4  24 

140 

9  20 

fe 

127* 

16 

4  0 

66 

16  30 

17 

1   8 

67 

4  28 

145 

9  40 

9 

135 

17 

4  15 

67 

16  45 

18 

1  18 

68 

4  32 

150 

10  0 

9* 

142* 

18 

4  30 

68 

17  0 

19 

1  16 

69 

4  36 

155 

10  20 

10 

150 

19 

4  45 

69 

17  15 

20 

1  20 

70 

4  40 

160 

1040 

10^ 

157* 

20 

5  0 

70 

17  30 

21 

1  24 

71 

4  44 

165 

11  0 

11 

165 

21 

5  15 

71 

17  45 

22 

1  28 

72 

4  48 

170 

11  20 

11* 

172* 

22 

5  30 

72 

18  0 

23 

1  32 

73 

4  52 

175 

11  40 

12 

180 

23 

5  45 

73 

18  15 

24 

1  36 

74 

4  56 

180 

12  0 

i  12* 

187* 

24 

6  0 

74 

18  30 

25 

1  40 

75 

5   0 

185 

12  20 

13 

195 

25 

6  15  ; 

To 

18  45 

26 

1  44 

76 

5   4 

190 

12  40 

13* 

202* 

26 

6  30  i 

76 

19  0 

27 

1  48 

77 

5   8 

195 

13  0 

14 

210 

27 

6  45  ! 

77 

19  15 

28 

1  52 

78 

5  12 

200 

13  20 

14* 

217* 

28 

7  0 

78 

19  30 

29 

1  56 

79 

5  16 

205 

13  40 

15 

225 

29 

7  15 

79 

19  45 

30 

2   0 

80 

5  20 

210 

14  0 

15| 

232* 

30 

7  30  i 

80 

20  0 

31 

2   4 

81 

5  24 

215 

14  20 

16 

240 

31 

745 

81 

20  15 

32 

2   8 

82 

5  28 

220 

14  40 

16* 

247* 

32 

8  0 

82 

•M  30 

33 

2  12 

83 

5  32 

225 

15  0 

17 

255 

33 

8  15 

83 

20  45 

34 

2  16 

84 

5  36 

230 

15  20 

17* 

262* 

34 

8  30 

84 

21  0 

35 

2  20 

85 

5  40 

235 

15  40 

18 

270 

35 

8  45 

85 

21  15 

36 

2  24 

86 

5  44 

240 

16  0 

;  18* 

277* 

36 

9  0 

86 

21  30 

37 

2  28 

8? 

5  48 

245 

16  20 

1  19 

285 

37 

9  15 

87 

21  45 

38 

2  32 

88 

5  52 

250 

16  40 

1  19* 

292i 

38 

9  30 

88 

22  0 

39 

2  36 

89 

5  56 

255 

17  0 

!  20 

300 

39 

9  45 

89 

22  15 

40 

2  40 

90 

6   0 

260 

17  20 

20i 

307* 

40 

10  0 

90 

22  30 

41 

2  44 

91 

6   4 

270 

18  0 

21 

315 

41 

10  15 

91 

22  45 

42 

2  48 

92 

6   8 

280 

18  40 

21* 

322* 

42 

10  30 

92 

23  0 

43 

2  52 

93 

6  12 

290 

19  20 

22 

330 

43 

10  45 

93 

23  15 

44 

2  56 

94 

6  16 

300 

20  0 

88* 

337* 

44 

11  0 

94 

23  30 

45 

3   0 

95 

6  20 

310 

20  40 

23 

345 

45 

11  15 

95 

23  45 

46 

3   4 

96 

6  24 

320 

21  20 

23* 

352* 

46 

11  30 

96 

24  0 

47 

3   8 

97 

6  28 

330 

22  0 

24 

360 

47 

11  45 

97 

24  15 

48 

3  12 

98 

6  32 

340 

22  40 

48 

12  0 

98 

24  30 

49 

3  16 

99 

6  36 

350 

23  20 

49 

12  15 

99 

24  45 

50 

3  20 

100 

6  40 

360   24  0 

50 

12  30 

100 

25  0 

358 


GENERAL  LIBRARY 
UNIVERSITY  OF  CALIFORNIA— BERKELEY 

RETURN  TO  DESK  FROM  WHICH  BORROWED 

This  book  is  due  on  the  last  date  stamped  below,  or  on  the 

date  to  which  renewed. 
Renewed  books  are  subject  to  immediate  recall. 


BNTERl 


RY  LOAN 


UNIV,  OF  CALIF,,  BERK. 


LiD  21-100m-l,'54(1837sl6)476 


01416 


